Category: Linear Algebra

The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns

Problem 295

Determine all possibilities for the number of solutions of each of the system of linear equations described below.

(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.

(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
 

(The Ohio State University, Linear Algebra Midterm Exam Problem)
Read solution

LoadingAdd to solve later

Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation

Problem 289

(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.

(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\] where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\] Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
Read solution

LoadingAdd to solve later

Summary: Possibilities for the Solution Set of a System of Linear Equations

Problem 288

In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.

Determine all possibilities for the solution set of the system of linear equations described below.

(a) A homogeneous system of $3$ equations in $5$ unknowns.

(b) A homogeneous system of $5$ equations in $4$ unknowns.

(c) A system of $5$ equations in $4$ unknowns.

(d) A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.

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

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

(g) A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.

(h) A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.

(i) A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.

(j) A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$.
 
Read solution

LoadingAdd to solve later

Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix

Problem 287

Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.

Determine which matrices are in the subspace $W$ and find the dimension of $W$.

(a) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &b &0 \\
0 & 0 & c
\end{bmatrix},\] where $a, b, c$ are distinct real numbers.

(b) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &a &0 \\
0 & 0 & b
\end{bmatrix},\] where $a, b$ are distinct real numbers.

 
Read solution

LoadingAdd to solve later

Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix

Problem 284

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.

(a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.

(b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?

(c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.

 
Read solution

LoadingAdd to solve later

Dual Vector Space and Dual Basis, Some Equality

Problem 282

Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.
Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that
\[x=\sum_{i=1}^nv^i(x)v_i\] for any vector $x\in V$.

 
Read solution

LoadingAdd to solve later

Quiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly Independent

Problem 281

(a) For what value(s) of $a$ is the following set $S$ linearly dependent?
\[ S=\left \{\,\begin{bmatrix}
1 \\
2 \\
3 \\
a
\end{bmatrix}, \begin{bmatrix}
a \\
0 \\
-1 \\
2
\end{bmatrix}, \begin{bmatrix}
0 \\
0 \\
a^2 \\
7
\end{bmatrix}, \begin{bmatrix}
1 \\
a \\
1 \\
1
\end{bmatrix}, \begin{bmatrix}
2 \\
-2 \\
3 \\
a^3
\end{bmatrix} \, \right\}.\]

(b) Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product
\[\mathbf{v}_i\cdot \mathbf{v}_j=0\] when $i\neq j$.
Prove that the set is linearly independent.

 
Read solution

LoadingAdd to solve later

Find a Nonsingular Matrix Satisfying Some Relation

Problem 280

Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\] where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
1 & 2 & -2
\end{bmatrix}\]

(b) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
2 & 1 & -4
\end{bmatrix}.\]

 
Read solution

LoadingAdd to solve later

Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent

Problem 279

Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
3 \\
1
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
4
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
2 \\
b
\end{bmatrix}.\]  
Read solution

LoadingAdd to solve later

Determine Linearly Independent or Linearly Dependent. Express as a Linear Combination

Problem 277

Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
\[\left\{\, \begin{bmatrix}
1 \\
0 \\
-1 \\
0
\end{bmatrix}, \begin{bmatrix}
1 \\
2 \\
3 \\
4
\end{bmatrix}, \begin{bmatrix}
-1 \\
-2 \\
0 \\
1
\end{bmatrix},
\begin{bmatrix}
-2 \\
-2 \\
7 \\
11
\end{bmatrix}\, \right\}.\]

 
Read solution

LoadingAdd to solve later

Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective

Problem 276

Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
\[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
a_0+a_2 & -a_0+a_3\\
a_1-a_2 & -a_1-a_3
\end{bmatrix}\] for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.

 
Read solution

LoadingAdd to solve later

The Inverse Matrix of an Upper Triangular Matrix with Variables

Problem 275

Let $A$ be the following $3\times 3$ upper triangular matrix.
\[A=\begin{bmatrix}
1 & x & y \\
0 &1 &z \\
0 & 0 & 1
\end{bmatrix},\] where $x, y, z$ are some real numbers.

Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.

 
Read solution

LoadingAdd to solve later

Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017.

Problem 273

(a) The given matrix is the augmented matrix for a system of linear equations.
Give the vector form for the general solution.
\[ \left[\begin{array}{rrrrr|r}
1 & 0 & -1 & 0 &-2 & 0 \\
0 & 1 & 2 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 \\
\end{array} \right].\]

(b) Let
\[A=\begin{bmatrix}
1 & 2 & 3 \\
4 &5 &6
\end{bmatrix}, B=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &0
\end{bmatrix}, C=\begin{bmatrix}
1 & 2\\
0& 6
\end{bmatrix}, \mathbf{v}=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}.\] Then compute and simplify the following expression.
\[\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.\]

 
Read solution

LoadingAdd to solve later

Prove a Given Subset is a Subspace and Find a Basis and Dimension

Problem 270

Let
\[A=\begin{bmatrix}
4 & 1\\
3& 2
\end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
\[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\]

(a) Prove that the subset $V$ is a subspace of $\R^2$.

(b) Find a basis for $V$ and determine the dimension of $V$.

 
Read solution

LoadingAdd to solve later