Solve this linear algebra multiple choice question.. ### Linear Algebra Problem: Vector in Span**Problem:**For what values of \( h \) is the vector \[ \begin{bmatrix}1 \\2 \\3 \\h^2 + 5 \end{bmatrix} \] in \[ \text{Span} \left\{ \begin{bmatrix}1 \\ 0 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix}1 \\ 1 \\ 1 \\ 2 \end{bmatrix} \right\}?\]**Explanation:**To determine for which values of \( h \) the vector lies in the span of the given set, we need to understand when the vector can be written as a linear combination of the given vectors.**Options for \( h \):**1. \( h \) can be any real number.2. \( h = -1 \)3. \( h = -1 \) and \( h = 1 \) *(Correct Answer)*4. \( h = 1 \)5. No such values for \( h \).The correct solution, \( h = -1 \) and \( h = 1 \), is highlighted in blue, indicating that these are the values of \( h \) that allow the vector to lie in the span of the given set of vectors.

Let us try to write h as a linear combination of vectors of given set.

Answered: For what values of h is the vector 4…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve this linear algebra multiple choice question..

### Linear Algebra Problem: Vector in Span

**Problem:**

For what values of \( h \) is the vector

\[
\begin{bmatrix}
1 \\
2 \\
3 \\
h^2 + 5
\end{bmatrix}
\]

in

\[
\text{Span} \left\{
\begin{bmatrix}
1 \\
0 \\
0 \\
2
\end{bmatrix},
\begin{bmatrix}
0 \\
1 \\
0 \\
2
\end{bmatrix},
\begin{bmatrix}
1 \\
1 \\
1 \\
2
\end{bmatrix}
\right\}?
\]

**Explanation:**

To determine for which values of \( h \) the vector lies in the span of the given set, we need to understand when the vector can be written as a linear combination of the given vectors.

**Options for \( h \):**

1. \( h \) can be any real number.
2. \( h = -1 \)
3. \( h = -1 \) and \( h = 1 \) *(Correct Answer)*
4. \( h = 1 \)
5. No such values for \( h \).

The correct solution, \( h = -1 \) and \( h = 1 \), is highlighted in blue, indicating that these are the values of \( h \) that allow the vector to lie in the span of the given set of vectors.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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