Karteikarten
Ohne Titel
Karten in diesem Stapel (10)
- 1What are the two Axioms for a Linear Transformation?Addition works
\[F(\overrightarrow{u}+\overrightarrow{v}) = F(\overrightarrow{u}) + F(\overrightarrow{v}) \]
Scalar multiplication works
\[ F(k\overrightarrow{u}) = k F(k\overrightarrow{u}) \] - 2For \(R^2\) what is the matrix for a shear of k along the x axis?\[ \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \]
- 3Given a transformation T, two basis B, B' and a vector \( \overrightarrow x \), what are the steps to construct T(\( \overrightarrow x \))? (Use the long way around)1. Construct \( T_B^{B\prime} \) by coordinatizeing the results of \( T(b_1) \) and \(T(b_2)\) into B'
2. coordinatizeing \(\overrightarrow x \) into B, \( [\overrightarrow x]_B \) (result in B)
3. Apply transformation \( T_B^{B\prime} [\overrightarrow x]_B \) (result in B')
4. Decoordinatize previous result \( T_B^{B\prime} [\overrightarrow x]_B \) by multiplying by the basis of B' - 4For \(R^2\) how does a shear of k along the x axis change the vector \(\langle x,y \rangle \) ?\[ T(\langle x,y \rangle) = \langle x+ky,y \rangle = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x +ky \\ y \end{bmatrix} \]
- 5Given a Basis B = \( \{ \langle a,c \rangle , \langle b,d \rangle \} \) and vector \( \overrightarrow v = \langle u_1,u_2 \rangle \), What is \( [\overrightarrow v]_B \) ?Set up system
\[ k_1 \cdot a + k_2 \cdot c = u_1 \]
\[ k_1 \cdot b + k_2 \cdot d = u_2 \]
Solve for \(k_1, k_2\)
\[ [\overrightarrow v]_B = \langle k_1,k_2 \rangle \] - 6For \(R^2\) what is the matrix for a compression/dilation of k along the x axis? y axis?\[ \begin{bmatrix} k & 0 \\ 0 & 1 \end{bmatrix} \]
\[ \begin{bmatrix} 1 & 0 \\ 0 & k \end{bmatrix} \] - 7For \(R^2\) what is the matrix for a flip across the x axis? y axis? the line y=x?\[ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \]
\[ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]
\[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \] - 8In general, what is a linear transformation from V to W equivalent to?Matrix multiplication
- 9What can any Linear Transformation be written as?The product of elementary matrices
- 10How does one deconstruct a linear transformation into elementary matrices?Perform Gaussian elimination to Identity, then reverse each step and interpret as an elementary matrix
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- Karteikarten am Ohne Titel, erstellt von Lewis Warne am 30/05/2018.
- Erstellt von
- Lewis Warne · Mai 2018
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- 30 Mai 2018
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