Q14E The vectors of the form ab0a聽聽... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q14E (page 164)

The vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ (where a and b are arbitrary real numbers) forms a subspace of $R^{4} .$

Short Answer

Expert verified

The above statement is true.

If W be the collection of the vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ , then W forms a subspace of $R^{4} .$

Step by step solution

Two Step Subspace Test

Let V be a vector space over the field K. Let W be a subset of V, then W be a subspace of V if and only if -

$0 鈭� W$
$u, v 鈭� W 鈬� u + v 鈭� W$
$u 鈭� W, a 鈭� F 鈬� a u 鈭� W$

To check whether the given form vectors form a subspace of R4.

Let W be the collection of the vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ in $R^{4} .$

Let $u, v 鈭� W$ then we have

$u = [\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}], v = [\begin{matrix} c \\ d \\ 0 \\ c \end{matrix}]$

Clearly, we see that W is non empty as

$[\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}] 鈭� W$

Now,

$u + V = [\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}] + [\begin{matrix} c \\ d \\ 0 \\ c \end{matrix}] = [\begin{matrix} a + c \\ b + d \\ 0 \\ a + c \end{matrix}] 鈭� W$

$鈬� u + v 鈭� W$

And for $纬鈭� F, and u 鈭� W$ , we have

$纬 u = 纬 [\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}] [\begin{matrix} 纬 a \\ 纬 b \\ 0 \\ 纬 a \end{matrix}] 鈭� W$

Final Answer

If, W be the collection of the vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ in $R^{4}$ androle="math" localid="1664262248842" $u, v 鈭� W$ then we have

$0 鈭� W$
$u, v 鈭� W 鈬� u + v 鈭� W$
$u 鈭� W, a 鈭� F 鈬� a u 鈭� W$

Since, subset W of $R^{4}$ satisfies all the above three conditions, thus W is a subspace of data-custom-editor="chemistry" $R^{4}$ .

Thus, the vectors of the form forms $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ the subspace of $R^{4}$ .

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91影视

The vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ (where a and b are arbitrary real numbers) forms a subspace of $R^{4} .$

Short Answer

Step by step solution

Two Step Subspace Test

To check whether the given form vectors form a subspace of R4.

Final Answer

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91影视

The vectors of the form ab0a(where a and b are arbitrary real numbers) forms a subspace ofR4.

Short Answer

Step by step solution

Two Step Subspace Test

To check whether the given form vectors form a subspace of R4.

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Mechanics Maths

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

The vectors of the form $[\begin{matrix} a \\ b \\ 0 \\ a \end{matrix}]$ (where a and b are arbitrary real numbers) forms a subspace of $R^{4} .$