Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.
Suppose that $S$ is an orthogonal set.

(a) Show that $S$ is linearly independent.

(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.

Proof.

(a) Show that $S$ is linearly independent.

Consider the linear combination
\[c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots +c_k \mathbf{v}_k=\mathbf{0}.\] Our goal is to show that $c_1=c_2=\cdots=c_k=0$.

We compute the dot product of $\mathbf{v}_i$ and the above linear combination for each $i=1, 2, \dots, k$:
\begin{align*}
0&=\mathbf{v}_i\cdot \mathbf{0}\\
&=\mathbf{v}_i \cdot (c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots +c_k \mathbf{v}_k)\\
&=c_1\mathbf{v}_i \cdot \mathbf{v}_1+c_2\mathbf{v}_i \cdot \mathbf{v}_2+\cdots +c_k \mathbf{v}_i \cdot\mathbf{v}_k.
\end{align*}

As $S$ is an orthogonal set, we have $\mathbf{v}_i\cdot \mathbf{v}_j=0$ if $i\neq j$.

Hence all terms but the $i$-th one are zero, and thus we have
\[0=c_i\mathbf{v}_i\cdot \mathbf{v}_i=c_i \|\mathbf{v}_i\|^2.\]

Since $\mathbf{v}_i$ is a nonzero vector, its length $\|\mathbf{v}_i\|$ is nonzero.
It follows that $c_i=0$.

As this computation holds for every $i=1, 2, \dots, k$, we conclude that $c_1=c_2=\cdots=c_k=0$.
Hence the set $S$ is linearly independent.

(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.

Suppose that $k=n$. Then by part (a), the set $S$ consists of $n$ linearly independent vectors in the dimension $n$ vector space $\R^n$.

Thus, $S$ is also a spanning set of $\R^n$, and hence $S$ is a basis for $\R^n$.

The post Orthogonal Nonzero Vectors Are Linearly Independent first appeared on Problems in Mathematics.

These are True or False problems.
For each of the following statements, determine if it contains a wrong information or not.

1. Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
2. The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
3. Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
4. The set
   \[\left\{\, \begin{bmatrix}
   1 \\
   0 \\
   0
   \end{bmatrix}, \begin{bmatrix}
   0 \\
   1 \\
   1
   \end{bmatrix} \,\right\}\] is an orthonormal set.

(Linear Algebra Exam Problem, The Ohio State University)

Solution.

(1) True or False? Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.

The answer is “False”. The definition of the range of the $5 \times 3$ matrix $A$ is
\[ \calR(A)=\{\mathbf{y}\in \R^5 \mid A\mathbf{x}=\mathbf{y} \text{ for some $\mathbf{x} \in \R^3$}\}.\] Note that to make sense the matrix product $A\mathbf{x}$, the size of the vector $\mathbf{x}$ must be $3$-dimensional because $A$ is $5\times 3$. Hence $\mathbf{y}=A\mathbf{x}$ is a $5$-dimensional vector, and thus the range $\calR(A)$ is a subspace of $\R^5$.
 

(2) True or False? The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.

The answer is “False”. The vector space $C[-1, 1]$ consists of all continuos functions defined on the interval $[-1, 1]$. Since $f(x)=x^2+1$ is a continuos function defined on $[-1, 1]$, it is in the vector space $C[-1, 1]$. The condition $f(0)=1\neq 0$ is irrelevant.
 

(3) True or False? Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.

The answer is “False”. First of all $\sin(x+y)\neq \sin(x)+\sin(y)$. For example, let $x=y=\pi/2$. Then
\[\sin\left(\,\frac{\pi}{2}+\frac{\pi}{2}\,\right)=\sin\left(\,\pi \,\right)=0\] and
\[\sin\left(\, \frac{\pi}{2} \,\right)+\sin\left(\, \frac{\pi}{2} \,\right)=1+1=2.\] Hence $\sin(x)$ is not a linear transformation.
 

(4) True or False? The given set is an orthonormal set.

The answer is “False”. The dot product of these vectors is
\[\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}\cdot \begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}=1\cdot 0+ 0\cdot 1 +0\cdot 1=0.\] Thus, the vectors are orthogonal. However the length of the second vector is
\[\sqrt{0^2+1^2+1^2}=\sqrt{2},\] hence it is not the unit vector.
So the set is orthogonal, but not orthonormal set.

Linear Algebra Midterm Exam 2 Problems and Solutions

• True of False Problems and Solutions (current problem): True or False problems of vector spaces and linear transformations
• Problem 1 and its solution: See (7) in the post “10 examples of subsets that are not subspaces of vector spaces”
• Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
• Problem 3 and its solution: Orthonormal basis of null space and row space
• Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
• Problem 5 and its solution: Determine value of linear transformation from $R^3$ to $R^2$
• Problem 6 and its solution: Rank and nullity of linear transformation from $R^3$ to $R^2$
• Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$
• Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

The post True or False Problems of Vector Spaces and Linear Transformations first appeared on Problems in Mathematics.