Tagged: kernel of a linear transformation

Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?

Problem 528

Let $V$ denote the vector space of all real $2\times 2$ matrices.
Suppose that the linear transformation from $V$ to $V$ is given as below.
2 & 3\\
5 & 7
2 & 3\\
5 & 7
\end{bmatrix}.\] Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

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Linear Transformation to 1-Dimensional Vector Space and Its Kernel

Problem 329

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.

(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.
Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then
\[B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}\] is a basis of $\R^n$.

(c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
\[\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}\] for some vector $\mathbf{v}\in \calN(T)$.

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Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

Problem 154

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}
x_1 \\
\end{bmatrix}\right )=\begin{bmatrix}
x_1-x_2 \\
x_1+x_2 \\

(a) Show that $T$ is a linear transformation.

(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

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