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Thread: A linear Transformation associated with matrix

  1. #1
    Member kjchauhan's Avatar
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    A linear Transformation associated with matrix

    Please Help me to solve these problems:

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

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

    3. If $\displaystyle \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$ is the matrix of a linear map $\displaystyle T:V_2 \to V_2$ relative to standard basis, then find the matrix $\displaystyle T^{-1}$ relative to the standard basis.
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  2. #2
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    Quote Originally Posted by kjchauhan View Post
    Please Help me to solve these problems:

    1. let A= $\displaystyle \begin{pmatrix} 1 & 1 & 2 & 3 \\ 1 & 0 & 1 & -1\\ 1 & 2 & 0 & 0 \end{pmatrix}$ and $\displaystyle B_1$ and $\displaystyle B_2$ are standard basis for $\displaystyle V_4$ and $\displaystyle V_3$. Determine the linear transformation $\displaystyle T:V_4 \to V_3$ respectively such that $\displaystyle A = (T:B_1,B_2)$.
    Do you understand what the question is asking? Suppose v= <w, x, y, z>. What is Av? That's all you need to say.

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

    3. If $\displaystyle \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$ is the matrix of a linear map $\displaystyle T:V_2 \to V_2$ relative to standard basis, then find the matrix $\displaystyle T^{-1}$ 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 $\displaystyle \theta$. The inverse of that is to rotate back. That's the same as rotating by what angle?
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    Accidental double post.
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