Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set
\[S=\{ \sqrt{x}, x^2 \}\]
in $C[3,10]$.
Show that the set $S$ is linearly independent in $C[3,10]$.
Note that the zero vector in $C[3,10]$ is the zero function $\theta(x)=0$.
Let us consider a linear combination
\[a_1\sqrt{x}+a_2x^2=\theta(x)=0.\]
We want to show that $a_1=a_2=0$.
Since this equality holds for any value of $x$ between $3$ and $10$, letting $x=4$ and $x=9$ yields the system of linear equations
\begin{align*}
2a_1+4a_2=0\\
3a_1+81a_2=0.
\end{align*}
Solving this system, we obtain $a_1=a_2=0$, and hence the set $S$ is linearly independent.
Comment.
We could have taken any two values between $3$ and $10$ for $x$ instead of $4$ and $9$, but the choice $x=4$ and $x=9$ made solving the system easier.
Also, we took two values for $x$ because we had two unknowns $a_1$, $a_2$, and thus we needed two equations to determine these unknown.
For a similar problem, show that the set $\{\sqrt{x}, x^2, x\}$ in the vector space $C[1, 10]$ is linearly independent.
In this case, there will be three unknowns that you want to show to be zero.
So you need to take three values for $x$.
A natural choice will be $x=1, 4, 9$.
Cosine and Sine Functions are Linearly Independent
Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$.
Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.
Proof.
Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]
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Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
-1
\end{bmatrix}, \begin{bmatrix}
-2 \\
1 \\
4
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Any Vector is a Linear Combination of Basis Vectors Uniquely
Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\]
where $c_1, c_2, c_3$ are […]
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Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let […]
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Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
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x_1 \\
x_2 \\
x_3
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in […]
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Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
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Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\]
be a matrix.
Find a basis of the null space of the matrix $A$.
(Remark: a null space is also called a kernel.)
Solution.
The null space $\calN(A)$ of the matrix $A$ is by […]