Category: Linear Algebra

Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space

Problem 131

Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
\[V:=\left\{ \quad\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \in \R^4
\quad \middle| \quad
x_1-x_2+x_3-x_4=0 \quad\right\}.\] Find a basis of the subspace $V$ and its dimension.

 
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Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$

Problem 125

Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers.
Determine whether $S$ is a subspace of $\R^3$.

 
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The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$

Problem 121

Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by
\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\] That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.

Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.
(Note that the null space is also called the kernel of $A$.)
 
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If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors?

Problem 120

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

 
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Express a Vector as a Linear Combination of Other Vectors


Problem 115

Express the vector $\mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}$ as a linear combination of the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix},
\mathbf{v}_2=
\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\mathbf{v}_3=
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}.\]

 
(The Ohio State University, Linear Algebra Exam)

 
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10 True or False Problems about Basic Matrix Operations

Problem 104

Test your understanding of basic properties of matrix operations.

There are 10 True or False Quiz Problems.

These 10 problems are very common and essential.
So make sure to understand these and don’t lose a point if any of these is your exam problems.
(These are actual exam problems at the Ohio State University.)

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

 
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Possibilities For the Number of Solutions for a Linear System

Problem 102

Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.


(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\] where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.


(b) $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.


(c) A homogeneous system of $3$ equations in $4$ unknowns.


(d) $A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:
\[\begin{bmatrix}
1 & 0 & -1 & 0 \\
0 &1 & 2 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.\] (The Ohio State University, Linear Algebra Exam)
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