Maiclubk
2021-02-11
Answered

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in $M}_{2\times 4$ with the property that FA = 0 (the zero matrix in ${M}_{3\times 4})$ . Determine if H is a subspace of $M}_{2\times 4$

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yagombyeR

Answered 2021-02-12
Author has **92** answers

A subspace is a subset H of a vector space V with the following 3 characteristics:

1. the zero vector of V is in H

2. the subspace gets sealed as more is added.

3. in the case of scalar multiplication, the subspace is closed.

1. Let A = 0. Then for any matrix F, FA = 0. Thus, $A\in H$. So H contains the zero vector.

2. Let ${A}_{1},{A}_{2}\in H$. Consider $A}_{1}+{A}_{2}.F({A}_{1}+{A}_{2})=F{A}_{1}4+F{A}_{2$ due to the distributive property of matrices. Since ${A}_{1},{A}_{2}\in H,F{A}_{1}+F{A}_{2}=0+0=0$.

Thus, ${A}_{1}+{A}_{2}\in H$, so H is closed under addition.

3. Let $c\in R,A\in H$. Consider cA. F(cA) = c(FA) according to scalar properties. Since $A\in H$, c(FA) =c(0) =0. Thus, $cA\in H$. So H is closed under scalar multiplication.

Thus, H fulfills all the requirements of the definition of a subspace of $M}_{2x4$.

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Let $\mathbb{K}$ be a field and, $a=\left(\begin{array}{l}{a}_{1}\\ {a}_{2}\\ {a}_{3}\end{array}\right),b=\left(\begin{array}{l}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right),c=\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right),d=\left(\begin{array}{c}{d}_{1}\\ {d}_{2}\\ {d}_{3}\end{array}\right)\in {\mathbb{K}}^{3}$

Show that a,b,c,d are in an affine plane if and only if

$det\left(\begin{array}{llll}{a}_{1}& {b}_{1}& {c}_{1}& {d}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}& {d}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}& {d}_{3}\\ 1& 1& 1& 1\end{array}\right)=0$

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Let $c,...,{c}_{r}$ some real numbers such that

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Let $c,...,{c}_{r}$ some real numbers such that

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Prove that all ${c}_{i}=0$

First it is more simple to work with the special case r=2, then since ${A}_{1}\cdot {A}_{2}=0$ Then there is no such ${c}_{1},{c}_{2}\in {\mathbb{R}}^{2}$ such that

${c}_{1}{A}_{1}+{c}_{2}{A}_{2}=0$

except for the case where ${c}_{1}={c}_{2}=0$

This last statement has to formalize in a more rigorous way, but i don’t how to do it i’ve just an intuition that this is the right case, and for higher dimensional vector this intuition would disappear.