A linear Transformation associated with matrix

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November 28th 2009, 12:45 AM
kjchauhan

A linear Transformation associated with matrix

Please Help me to solve these problems:

1. let A= $\begin{pmatrix} 1 & 1 & 2 & 3 \\ 1 & 0 & 1 & -1\\ 1 & 2 & 0 & 0 \end{pmatrix}$ and $B_1$ and $B_2$ are standard basis for $V_4$ and $V_3$ . Determine the linear transformation $T:V_4 \to V_3$ respectively such that $A = (T:B_1,B_2)$ .

2. let A= $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B_1 = \{(1,1,1),(1,0,0),(0,1,0)\}$ and $B_2=\{(1,2,3),(1,-1,1),(2,1,1)\}$ . Determine the linear transformation $T:V_3 \to V_3$ respectively such that $A = (T:B_1,B_2)$ .

3. If $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$ is the matrix of a linear map $T:V_2 \to V_2$ relative to standard basis, then find the matrix $T^{-1}$ relative to the standard basis.
November 28th 2009, 04:29 AM
HallsofIvy

Quote:

Originally Posted by kjchauhan

Please Help me to solve these problems:

1. let A= $\begin{pmatrix} 1 & 1 & 2 & 3 \\ 1 & 0 & 1 & -1\\ 1 & 2 & 0 & 0 \end{pmatrix}$ and $B_1$ and $B_2$ are standard basis for $V_4$ and $V_3$ . Determine the linear transformation $T:V_4 \to V_3$ respectively such that $A = (T:B_1,B_2)$ .

Do you understand what the question is asking? Suppose v= <w, x, y, z>. What is Av? That's all you need to say.

Quote:

2. let A= $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B_1 = \{(1,1,1),(1,0,0),(0,1,0)\}$ and $B_2=\{(1,2,3),(1,-1,1),(2,1,1)\}$ . Determine the linear transformation $T:V_3 \to V_3$ respectively such that $A = (T:B_1,B_2)$ .

This says, then that A(1,1,1)= 1(1,1,1)+ 1(1,0,0)+ 1(0,1,0), A(1,0,0)= 1(1,1,1)+ 0(1,0,0)+ 0(0,1,0), and A(0,1,0)= 0(1,1,1)+ 1(1,0,0)+ 0(0,1,0). How do you "describe" that linear transformation? What is A(x,y,z)?

Quote:

3. If $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$ is the matrix of a linear map $T:V_2 \to V_2$ relative to standard basis, then find the matrix $T^{-1}$ relative to the standard basis.

There are two ways to do this.
One: find the inverse matrix.
Two: recognize that this matrix rotates every point by an angle $\theta$ . The inverse of that is to rotate back. That's the same as rotating by what angle?
November 28th 2009, 04:40 AM
HallsofIvy

Accidental double post.