orthogonal set | Problems in Mathematics

10/24/2017

Orthogonal Nonzero Vectors Are Linearly Independent

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 $\R^n$.

04/06/2017

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

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