Recent questions tagged annotated-8

Say that $S: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ and $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ are linear transformations. Which of the following must be true...

Let $T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$ be a linear transformation with matrix $A$.How many rows and columns does $A$ have?If $x$ is in $\mathrm{R}^{n}$, then ...

Circle $\mathbf{T}$ if the statement is always true, and circle $\mathbf{F}$ otherwise.A. $\mathbf{T} \quad \mathbf{F} \quad$ If $T: \mathbf{R}^{n} \rightarrow \mathbf{R}...

Consider $T: \mathbf{R}^{3} \rightarrow \mathbf{R}^{4}$ given by$$T(x, y, z)=(x, x+z, 3 x-4 y+z, x) .$$Is $T$ one-to-one? Justify your answer.

Which of the following statements are true?The transformation $T(x, y, z)=(y-1,4 x+z, x)$ is an onto linear transformation.Let $A$ be a $3 \times 4$ matrix. Then, the tra...

Which of the following is not a condition for a transformation $T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$ with standard matrix $A$ to be onto?The columns of $A$ span ...

If $T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$ is onto, what can we say about the relative sizes of $n$ and $m$ ?

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the matrix mapping with corresponding matrix$$\mathbf{A}=\left[\begin{array}{ccc}1 & 2 & -1 \\-3 & -4 & 2 \\5 & 2 & ...

Let $\mathrm{T}_{\mathrm{A}}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3}$ be the matrix mapping with corresponding matrix$$\mathbf{A}=\left[\begin{array}{cccc}1 & 2 & -1 &...

If $\mathrm{T}_{\mathbf{A}}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a matrix mapping corresponding to $\mathbf{A} \in M_{m \times n}$ and $n<m$ then $T_A$ cannot b...

A vector mapping $\mathrm{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is said to be onto if for each $\mathbf{b} \in \mathbb{R}^{m}$ there is at least one $\mathbf{x} ...

Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation and let $A$ be the standard matrix for $T$. Then:a. $T$ maps $\mathbb{R}^{n}$ onto $\mathbb{...

I.A vector mapping $\mathrm{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is said to be one-to-one if for each $\mathbf{b} \in \operatorname{Range}(\mathrm{T})$ there ex...

Let $\mathrm{T}_{\mathrm{A}}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be the matrix mapping with corresponding matrix$$\mathbf{A}=\left[\begin{array}{cc}1 & 4 \\-3 & 2...

Suppose $T: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ is a linear transformation whose range is a plane, and suppose$$T\left(\begin{array}{l}0 \\1 \\0\end{array}\right)=...

Let $\mathrm{T}_{\mathrm{A}}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3}$ be the matrix mapping with matrix$$\mathbf{A}=\left[\begin{array}{cccc}3 & -2 & 6 & 4 \\-1 & 0 & ...

Let $\mathrm{T}_{\mathrm{A}}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be the matrix mapping with matrix$$\mathbf{A}=\left[\begin{array}{cc}1 & 0 \\3 & -1 \\2 & 0\end{a...

Let $T: \mathbf{R}^{\mathbf{2}} \rightarrow \mathbf{R}^{2}$ be the linear transformation satisfying$$T\binom{1}{0}=\binom{1}{2} \quad \text { and } \quad T\binom{1}{1}=\b...