| layout | title | order | permalink |
|---|---|---|---|
page |
Oceanverse |
5 |
/linalg75/ |
- TOC {:toc}
-
Consider the matrix $$ \left[ \begin{array}{cc} 1 & 2\ 3 & 4 \end{array}\right] $$
- What does it remind you of?
- What does it denote?
- Where and why do we use a matrix?
- Can you enlist a few applications of matrices?
-
Define a function. What is a surjective, injective and bijective function?
-
Give an example of a function $$ f:\mathbb{R}^2\rightarrow\mathbb{R}^2 $$
-
Give an example of a very nice function
$$f:\mathbb{R}^2\rightarrow\mathbb{R}^2$$ - Make extra efforts to make this function really nice.
- Explain what is so nice about your function?
- Why should one study such functions?
-
Define a function $$ \phi : \mathbb{R}^2 \rightarrow \mathbb{R}^2
$$, which satisfies the following property: The point $$ \phi(2,3)=(7,4)$$. Note that this function should be defined at all points on $$ \mathbb{R}^2$$.- What are the properties of your function? Observe it microscopically.
-
Given a function
$$\phi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ which satisfies the following two conditions:$$\phi(u+v)=\phi(u) +\phi(v)$$ $$\phi(\alpha v)=\alpha\phi(v)$$ where
$$u=( x_{1} , y_{1} )$$ ,$$v=( x_{2} , y_{2} )$$ and$$u,v \in \mathbb{R}^2$$ - What can one say of such functions?
- Observe this function closely, such functions form the crux of several discplines, mainly of sciences and engineering. Spend at least 20 minutes dissecting them to the best possible extent.
-
Consider
$$\mathbb{R}^2$$ what are all the properties of this set?$$\mathbb{R}^2$$ is called a space of all vectors, aka a vector space. Lookup for the definition of a vector space. -
A subset of a vector space which in itself is a vector space is caled a subspace. Give an example of a subspace of
$$\mathbb{R}^2$$ . -
Given a vector
$$(1,7)$$ , what does the set$${ \alpha(1,7) | \alpha \in \mathbb{R}}$$ represent? Is it a subspace of$$\mathbb{R}^2$$ ? -
Is
$$\mathbb{R}^3$$ a vector space? -
Conside the two points
$$(1,2,3)$$ and$$(4,5,7)\in \mathbb{R}^3$$ . What does the following set denote:$${\alpha(1,2,3)+\beta(4,5,7) | \alpha, \beta \in \mathbb{R}}$$ . Is this a subspace? -
Consider a straight line
$$y=2x+1$$ in$$\mathbb{R}^2$$ , does it form a subspace of$$\mathbb{R}^2$$ ? -
Consider a unit circle in
$$\mathbb{R}^2$$ , centered at origin, is it a subspace of$$\mathbb{R}^2$$ ? -
What are all the subspaces of
$$\mathbb{R}^2$$ ? -
What are all the subspaces of
$$\mathbb{R}^3$$ ? -
Given
$$\mathbb{R}^3$$ , pick any two points$$u,v\in \mathbb{R}^3$$ . Note that$${\alpha u+\beta v | \alpha, \beta \in \mathbb{R}}$$ is a subspace of$$\mathbb{R}^3$$ . Generalize this idea! -
The set
$${\alpha \vec{u} + \beta \vec{v} | \alpha, \beta \in \mathbb{R}}$$ is called the linear combination of vectors$$\vec{u}$$ and$$\vec{v}$$ . We can generalize this to$$k$$ vectors. Observe what this set is all about? -
We say that a vector
$$\vec{w}$$ is manufactured by$$\vec{u}$$ and$$\vec{v}$$ if$$w\in {\alpha \vec{u} +\beta \vec{v} | \alpha, \beta \in \mathbb{R} }$$ . -
Show that
$$(1,2,3)$$ and$$(4,5,6)$$ can manufacture$$(7,8,9)$$ . Also$$(4,5,6)$$ and$$(7,8,9)$$ can manufacture$$(1,2,3)$$ . Finally$$(4,5,6)$$ can be manufactured by the other two vectors. -
Can
$$(2,1,0)$$ and$$(3,0,8)$$ manufacture$$(1,1,1)$$ ? -
Can
$$(0,0,1)$$ and$$(0,1,0)$$ manufacture$$(1,0,0)$$ ? -
When can two vectors in
$$\mathbb{R}^3$$ manufacture a given third vector? -
When can two vectors in
$$\mathbb{R}^3$$ fail to manufacture a given third vector? -
If
$${\vec{u},\vec{v},\vec{w}}$$ are such that a vector in this set can be manufactured by some vectors in the same set, then such a set is called a linearly dependent set. -
Give examples of a linearly dependent set in
$$\mathbb{R}^3$$ and get conversant with the definition. -
If
$${\vec{u},\vec{v},\vec{w}}$$ are such that no vector in this set can be manufactured by any combination of vectors in the same set, then such a set is called a linearly independent set. -
Give examples of a linearly dependent set in
$$\mathbb{R}^3$$ and$$\mathbb{R}^2$$ and get conversant with the definition. -
Are
$$(1,2), (3,4)$$ linearly independent in$$\mathbb{R}^2$$ ? Prove! -
Are
$$(1,1),(2,3),(7,17)$$ Linearly Independent or Dependent? -
-
Construct a set of 3 vectors that are Linearly Independent in
$$\mathbb{R}^2$$ . Can you? -
Construct a set of 3 vectors that are Linearly Independent in
$$\mathbb{R}^3$$ . Can you? -
Construct a set of
$$k$$ vectors that are Linearly Independent in$$\mathbb{R}^k$$ . Can you? -
Construct a set of
$$k+1$$ vectors that are Linearly Independent in$$\mathbb{R}^k$$ . Can you?
-
-
-
Show that in
$$\mathbb{R}^2$$ we can at most have 2 Linearly Independent vectors. -
Show that in
$$\mathbb{R}^3$$ we can at most have 3 Linearly Independent vectors. -
Show that in
$$\mathbb{R}^n$$ we can at most have$$n$$ Linearly Independent vectors.
-
-
Set of any two linearly independent vectors in
$$\mathbb{R}^2$$ is called a basis. Set of any three linearly independent vectors in$$\mathbb{R}^3$$ is called a basis. Similarly, for$$n$$ . -
Show that a basis can manufacture any vector of the vector space.
-
Show that any set of linearly independent vectors form a subset of some basis. In other words, one can include a few more elements to a linearly independent set and make it a basis.
-
Show that the number of elements in any basis for a given vector space is always constant.
-
Try taking a few vectors in
$$\mathbb{R}^3$$ and discuss its linear independence or linear dependence. Try atleast 10 examples and familiarize yourself. -
Show that any 3 vectors on a plane passing through the origin in
$$\mathbb{R}^3$$ cannot be linearly independent. Prove. -
What are all the subspaces of
$$R^3$$ . -
Consider a subspace of
$$R^3$$ and write down its basis. Do this for 3 to 4 different subspaces. -
Give an example of a function
$$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$$ such that the range of the function is the straight line$$y=10x$$ . -
As an example consider the following basis set of
$$\mathbb{R}^2 : \left {(1,2),(2,2)\right }$$ . Construct a function$$\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ such that:$$\phi(\vec{u}+\vec{v})=\phi(\vec{u})+\phi(\vec{v})$$ $$\phi(\alpha \vec{u})= \alpha \phi(\vec{u})$$ Such a function is called a LT function,
-
Where is (1,2) and (2,2) mapped?
-
Where is (1,0) and (0,1) mapped?
-
What is the range of this function?
-
-
Consider a function
$$\phi$$ which is LT which maps$$\phi(1,2)=(3,4)$$ $$\phi(6,6)= (2,0)$$ -
How many such functions can you construct?
-
What is the range of this function?
-
-
Discuss the range of the following functions, given their values at a few points:
$$\phi(1,2)=(3,4)$$ $$&$$ $$\phi(1,1)=(6,8)$$ $$\phi(1,1)=(2,2)$$ $$&$$ $$\phi(3,3)=(4,4)$$ $$\phi(1,2)=(3,5)$$ $$&$$ $$\phi(0,8)=(3,0)$$ $$\phi(-2,2)=(0,0)$$ $$&$$ $$\phi(8,2)=(1,1)$$ $$\phi(1,1)=(2,2)$$ $$&$$ $$\phi(1,2)=(3,3)$$ -
Give an example of an LT function which maps a L.I. set of vectors to a L.I. set of vectors i.e.
$$\phi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$$ , where$$\left { \vec{u},\vec{v} \right }$$ are L.I.$$&$$ $$\left {\phi(\vec{u}), \phi(\vec{v})\right }$$ are also L.I. -
Same as the previous question, but
$$\left { \vec{u},\vec{v} \right }$$ is L.I. and$$\left {\phi(\vec{u}), \phi(\vec{v})\right }$$ is L.D.
What can one say about the range? -
Note that range of a L.I function is always a subspace.
-
Note that the set N=$${\vec{v}: \phi(\vec{v})=0}$$ is a subspace of
$$R^2$$ where$$\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ is L.I . Take atleast 5 different examples of L.I functions and see if this is true. The set$$N\subseteq Domain$$ is called the null space of$$\phi$$ . -
Construct a Linear Transformation
$$\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ ,$$\phi(0,1)=(3,4)$$ and$$\phi(1,0)=(1,2)$$ . Construct an inverse of linear transformation and observe it carefully. Is it a L.T function too? -
Do you realize the importance of inverting a matrix?
-
L.T stands for Linear Transformation .It is aka a matrix.
-
Define a L.T.
$$\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$$ such that$$\phi(0,1)=(2,3)$$ and$$\phi(1,0)=(7,4)$$ . This function$$\phi$$ is same as the matrix $$\begin{bmatrix} 7 & 2 \ 4 & 3 \end{bmatrix}$$.
$$\phi(x,y)$$ is same as $$\begin{bmatrix} 7 & 2 \ 4 & 3 \end{bmatrix}$$ $$\begin{bmatrix} x\y \end{bmatrix}$$, Verify and make your share of observations.
What is$$\phi^{-1}$$ ? Is this the traditional matrix inverse?
Think ! -
Consider a L.T.
$$\phi$$ such that :$$\phi(1,2)=(3,5)$$ and$$\phi(3,1)=(8,2)$$ . What is the matrix equivalent of$$\phi$$ ? -
Solve the following :
-
$$\begin{array} 3x-2y=15 \ x+4y=19 \end{array}$$
-
Isn't this the same as $$\begin{bmatrix} 1 & -2 \ 1 & 4 \end{bmatrix}$$ $$\begin{bmatrix} x\y \end{bmatrix}$$ = $$\begin{bmatrix} 15\19 \end{bmatrix}$$
-
You are trying to find if there is an element in the domain which maps to (15,19) .
-
what exactly is happening here ? (Say all that you can)
-
-
Given $$\overrightarrow{V}{1}$$ , $$\overrightarrow{V}{2}$$ &
$$\overrightarrow{V}_{3}$$ are L.I. , S.T.-
$$\left { \overrightarrow{V}{2} - \overrightarrow{V}{3} , \overrightarrow{V}{1} - \overrightarrow{V}{3} , \overrightarrow{V}{1} - \overrightarrow{V}{2} \right }$$ are L.D.
-
$$\left { \overrightarrow{V}{1} + \overrightarrow{V}{2} , \overrightarrow{V}{1} + \overrightarrow{V}{3} , \overrightarrow{V}{2} + \overrightarrow{V}{3} \right }$$ are L.I.
-
-
Given vectors $$\left { \overrightarrow{V}{1} , \overrightarrow{V}{2} , \overrightarrow{V}{3} , \overrightarrow{V}{4} \right }$$ their sum is 0 = $$\overrightarrow{V}{1}$$ + $$\overrightarrow{V}{2}$$ + $$\overrightarrow{V}{3}$$ + $$\overrightarrow{V}{4}$$ . Is this set L.I or L.D. ?
-
Show that the following are equivalent:
-
The vectors
$${\vec{v}_1,\vec{v}_2,\dots, \vec{v}_n}$$ are linearly independent. -
$$\forall \alpha_1, \alpha_2, \dots, \alpha_n \in \mathbb{R}\ \left( \sum_{i=1}^{n} \alpha_i \vec{v}_i = 0 \implies \alpha_j = 0\ \forall 1 \leq j \leq n\right)$$ .
-
-
In the following six matrices find out the following:
-
Rank of the matrix
-
Dimension of the range
-
$$\left[ \begin{array}{cc} 1 & 2\ 2 & 4 \end{array}\right]$$
-
$$\left[ \begin{array}{cc} 1 & 2\ 3 & 4 \end{array}\right]$$
-
$$\left[ \begin{array}{cc} 7 & -7\ 2 & -2 \end{array}\right]$$
-
$$\left[ \begin{array}{cc} 0 & 1\ 1 & 0 \end{array}\right]$$
-
$$\left[ \begin{array}{cc} 1 & 1\ 3 & 3 \end{array}\right]$$
-
$$\left[ \begin{array}{ccc} 1 & 2 & 3\ 4 & 5 & 6 \ 7 & 8 & 9 \end{array}\right]$$
-
What is happening here? Describe in detail.
-
-
The dimension of the range of the matrix
$$M$$ and$$M^{T}$$ is always the same. Why? -
Take three linearly independent vectors in
$$\mathbb{R}^3$$ . Show that they form a basis of$$\mathbb{R}^3$$ . -
Consider the matrix $$\left[ \begin{array}{ccc} 1 & 4 & 7\ 2 & 5 & 8 \ 3 & 6 & 9 \end{array}\right]$$. The dimension of the range is 2. The range of this linear transformation is obviously a linear combination of three vectors. Do you see which are those three vectors?
-
Consider any
$$2 \times 2$$ matrix. Do you observe that the :Dimension of the Null space & the Dimension of the range are in someway related?
-
What about a
$$3 \times 3$$ matrix? -
Can you generalise this to an
$$n\times n$$ matrix? -
When is a L.T. function one-one and when is it onto?
-
A
$$2\times 2$$ matrix A can be seen as two vectors placed as columns. For example$$(1,2)$$ and$$(3,4)$$ when placed as columns give rise to the matrix $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$.Note that $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$ $$\begin{bmatrix} \alpha \ \beta \end{bmatrix}$$
$$\in$$ $${{ \alpha(1,2)+\beta(3,4) : \alpha, \beta \in \mathbb{R} } }$$ .Note that matrix, $$\begin{bmatrix} a & b \ c & d \end{bmatrix}$$ $$\begin{bmatrix} x \ y \end{bmatrix}$$ simply is
$$x(a,c)+y(b,d)$$ .Note that a matrix takes a vector
$$(x,y)$$ to a linear combination of the column vectors.It is now clear that the range of the matrix, say$$\begin{bmatrix} 1 & 1 & 4 \ 0 & 2 & 4 \ 1 & 3 & 4 \end{bmatrix}$$ is nothing but the linear combination of the vectors
$$(1,0,1), (1,2,3), (4,4,4)$$ . -
Consider a
$$10\times 10$$ matrix$$A$$ , defined as following:$$A[i,j] = 0$$ if$$i+j\cong 0(mod2)$$ $$A[i,j] = 1$$ if$$i+j\cong 1(mod2)$$ What is the dimension of the range?
-
Consider the sub-space
$$S:(y=13x)$$ of$$\mathbb{R}^2$$ -
Give an example of a
$$\phi$$ such that$$S$$ is its null space. -
Give an example of a
$$\phi$$ such that$$S$$ is the the range of$$\phi$$ .
-
-
Let
$$A =$$ $$\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$$-
Show that the null space of
$$A \subset$$ Null space of$$A^2$$ -
Is this true for any
$$2\times 2$$ matrix. -
Is this true for any
$$3\times3$$ matrix. -
Is this true for any
$$n\times n$$ matrix.
-
-
Define column space. S.T. Column space of
$$A^2 \subset$$ Column space of$$A$$ . -
Solve the following simultaneous equation:
-
$$ \begin{array} _x + 4y + 7z =9\ 2x + 5y + 8z = 9\ 3x +6y + 9z = 9 \end{array} $$
-
What is the null space of $$\begin{bmatrix} 1 & 4 & 7 \ 2 & 5 & 8 \ 3 & 6 & 9 \end{bmatrix}$$
-
The above two questions are strongly related and make a theory. What is your obseration?
-
-
What does one mean by the linear combination of vectors and what is a linear span?
-
Show that, if a matrix
$$A$$ has linearly independent column vectors, then the columns of the matrix$$A^2$$ are also linearly independent. -
$$A\vec{x} = A\vec{y} \iff \vec{x}=\vec{y}$$ . Is this true? If False, when is the statement False and what leads to the falsity of the statement? -
Let
$$M$$ be a$$3\times 3$$ matrix, such that, the dim(Range) =3. Show that,$$M^2 = M \iff M$$ is an identity matrix. -
Consider a
$$10 \times 10$$ matrix with all its entries to be$$1$$ . What is the dimension of its range?