# Matrix transformation questions and answers

Recent questions in Matrix transformations
Matrix transformations

### For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. $$A=\begin{bmatrix}2&3&5&-9\\-8&-9&-11&21\\4&-3&-17&27\end{bmatrix}$$ Find a nonzero vector in Nul A. $$A=\begin{bmatrix}-3\\2\\0\\1\end{bmatrix}$$

Matrix transformations

### The matrix in Hessenberg form with the help of similarity transformation and the matrix in the similarity transformations.

Matrix transformations

### Determine if the columns of the matrix form a linearly independent set. Justify each answer. $$\begin{bmatrix}1&4&-3&0\\-2&-7&5&1\\-4&-5&7&5\end{bmatrix}$$

Matrix transformations

### With T defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. $$A\begin{bmatrix}1&-5&-7\\-3&7&5\end{bmatrix},B=\begin{bmatrix}-2\\-2\end{bmatrix}$$

Matrix transformations

### Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. $$\begin{bmatrix}1 & 3 & -2 \\-4 & h & 8 \end{bmatrix}$$

Matrix transformations

### Assume that T is a linear transformation. Find the standard matrix of T. $$\displaystyle{T}:{\mathbb{{{R}^{{{2}}}}}}\rightarrow{\mathbb{{{R}^{{{2}}}}}}$$ is a vertical shear transformation that maps $$e_{1}\ into\ e_{1}-2e_{2}$$ but leaves the vector $$\displaystyle{e}_{{{2}}}$$ unchanged.

Matrix transformations

### Let W be the subspace spanned by the given vectors. Find a basis for $$w_1=\begin{bmatrix}1\\-1\\3\\-2\end{bmatrix},w_2=\begin{bmatrix}0\\1\\-2\\1\end{bmatrix}$$

Matrix transformations

### Find the area of the parallelogram whose vertices are listed. (0,0), (5,2), (6,4), (11,6)

Matrix transformations

### The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y;x,y; or x, y, z;x,y,z; or $$\displaystyle{x}_{{{1}}},{x}_{{{2}}},{x}_{{{3}}},{x}_{{{4}}}$$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $$\begin{bmatrix}1&0&0&0&|&1 \\0&1&0&0&|&2 \\0&0&1&2&|&3 \end{bmatrix}$$

Matrix transformations

### Find an orthogonal basis for the column space of each matrix. $$\begin{bmatrix}3&-5&1\\1&1&1\\-1&5&-2\\3&-7&-8\end{bmatrix}$$

Matrix transformations

### Consider the linear system $$\vec{y}'=\begin{bmatrix}-3 & -2 \\ 6 & 4 \end{bmatrix}\vec{y}$$ a) Find the eigenvalues and eigenvectors for the coefficient matrix $$\lambda_{1}1,\ \vec{v}_{1}=\begin{bmatrix} -1 \\ 2 \end{bmatrix}$$ and $$\lambda_{2}=0,\ \vec{v}_{2}=\begin{bmatrix} -2 \\ 3 \end{bmatrix}$$ b) For each eigenpair in the previos part, form a solution of $$\displaystyle\vec{{{y}}}'={A}\vec{{{y}}}$$. Use t as the independent variable in your answers. $$\vec{y}_{1}(t)=\begin{bmatrix} & \\ & \end{bmatrix}$$ and $$\vec{y}_{2}(t)=\begin{bmatrix} -2 \\ 3 \end{bmatrix}$$ c) Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solution? No, it is not a fundamental set.

Matrix transformations

### Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. $$\left[\begin{array}{c} 7 & -3 \\ 2 & 1 \\ 9 & -6 \\ -3 & 2\end{array}\right] \left[\begin{array}{c} -2 \\ -5\end{array}\right]=\left[ \begin{array}{c}1 \\ -9 \\ 12 \\ -4\end{array}\right]$$

Matrix transformations

### Consider the $$\displaystyle{3}\times{3}$$ matrices with real entrices. Show that the matrix forms a vector space over R with respect to matrix addition and matrix multiplication by scalars?

Matrix transformations

### Find an explicit description of Nul A by listing vectors that span the null space. $$A=\begin{bmatrix}1&5&-4&-3&1\\0&1&-2&1&0\\0&0&0&0&0\\\end{bmatrix}$$

Matrix transformations

### Соnstruct a nonzerо $$\displaystyle{3}\times{3}$$ matriх А and a nonzerа vector b such that b is in Nul A.

Matrix transformations

### Find k such that the following matrix M is singular. $$M=\begin{bmatrix}-1 & -1 & -2\\ 0 & -1 & -4 \\ -12+k & -2 & -2 \end{bmatrix}$$ $$\displaystyle{k}=?$$

Matrix transformations

### Let A be an $$\displaystyle{m}\times{n}$$ matrix, and $$\displaystyle{C}={A}{B}$$. Show that: a) $$Null(B)$$ is a subspace of $$Null(C)$$. b) $$Null(C)^{\bot}$$ is a subspace of $$Null(B)^{\bot}$$ and, consequently, $$\displaystyle{C}{o}{l}{\left({C}{T}\right)}$$ is a subspace of $$\displaystyle{C}{o}{l}{\left({B}{T}\right)}$$.

Matrix transformations

### Assume that T is a linear transformation. Find the standard matrix of T. $$\displaystyle{T}:{\mathbb{{{R}}}}^{{{2}}}\rightarrow{\mathbb{{{R}}}}^{{{4}}},{T}{\left({e}_{{{1}}}\right)}={\left({3},{1},{3},{1}\right)}\ {\quad\text{and}\quad}\ {T}{\left({e}_{{{2}}}\right)}={\left(-{5},{2},{0},{0}\right)},\ {w}{h}{e}{r}{e}\ {e}_{{{1}}}={\left({1},{0}\right)}\ {\quad\text{and}\quad}\ {e}_{{{2}}}={\left({0},{1}\right)}$$.

Matrix transformations