/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Let \(\mathbf{a}_{1}\) and \(\ma... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 9

Let \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\) be linearly independent vectors in \(\mathbb{R}^{3}\) and let \(\mathbf{x}\) be a vector in \(\mathbb{R}^{2}\) (a) Describe geometrically \(\operatorname{Span}\left(\mathbf{a}_{1}, \mathbf{a}_{2}\right)\) (b) If \(A=\left(\mathbf{a}_{1}, \mathbf{a}_{2}\right)\) and \(\mathbf{b}=A \mathbf{x},\) then what is the dimension of \(\operatorname{Span}\left(\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{b}\right) ?\) Explain.

Short Answer

Expert verified
(a) The Span(\(饾悮鈧�,饾悮鈧俓)) forms a plane in 鈩澛�. (b) The dimension of Span(\(饾悮鈧�,饾悮鈧�,饾悰\)) is 2, as the addition of 饾悰 to the span does not increase the dimension since it also lies within the plane created by Span(\(饾悮鈧�,饾悮鈧俓)).

Step by step solution

01

Understand the concept of Span and its geometrical representation

The span of a set of vectors is the set of all linear combinations of those vectors. In 鈩澛�, the span of two linearly independent vectors will form a plane. Since the given vectors 饾悮鈧� and 饾悮鈧� are linearly independent, they do not lie on the same line, and their linear combinations will create a plane in 鈩澛�. So, the answer to (a) is that the Span(饾悮鈧�,饾悮鈧�) forms a plane in 鈩澛�.
02

Analyze the relationship between Span(饾悮鈧�,饾悮鈧�,饾悰) and Span(饾悮鈧�,饾悮鈧�) using the given matrices A and b

We know that the vector 饾悰 = A 饾惐, which means that 饾悰 is formed by the linear combination of 饾悮鈧� and 饾悮鈧�. 饾悰 = 饾悮鈧� * 饾懃鈧� + 饾悮鈧� * 饾懃鈧�, where 饾懃鈧� and 饾懃鈧� are components of vector 饾惐 in 鈩澛�. Since 饾悰 is a linear combination of 饾悮鈧� and 饾悮鈧�, it must also lie within the plane created by Span(饾悮鈧�,饾悮鈧�).
03

Find the dimension of Span(饾悮鈧�,饾悮鈧�,饾悰)

As we proved in step 2 that 饾悰 lies in the plane formed by Span(饾悮鈧�,饾悮鈧�), adding this vector to the span will not change the dimension of the subspace created by 饾悮鈧� and 饾悮鈧�. The dimension of the span of two linearly independent vectors in 鈩澛� is 2 (because it forms a plane), so the dimension of Span(饾悮鈧�,饾悮鈧�,饾悰) is also 2. In conclusion, the answer to (b) is that the dimension of Span(饾悮鈧�,饾悮鈧�,饾悰) is 2, as the addition of 饾悰 to the span does not increase the dimension since it also lies within the plane created by Span(饾悮鈧�,饾悮鈧�).

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Most popular questions from this chapter

Let \(S, T,\) and \(U\) be subspaces of a vector space \(V\) We can form new subspaces by using the operations of \(\cap\) and \(+\) defined in Exercises 20 and \(22 .\) When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that \\[ a(b+c)=a b+a c \\] It is natural to ask whether similar distributive laws hold for the two operations with subspaces. (a) Does the intersection operation for subspaces distribute over the addition operation? That is does \\[ S \cap(T+U)=(S \cap T)+(S \cap U) \\] (b) Does the addition operation for subspaces disdoes tribute over the intersection operation? That is, \\[ S+(T \cap U)=(S+T) \cap(S+U) \\]

Determine whether the following are subspaces of \(P_{4}(\text { be careful! })\) (a) The set of polynomials in \(P_{4}\) of even degree (b) The set of all polynomials of degree 3 (c) The set of all polynomials \(p(x)\) in \(P_{4}\) such that \(p(0)=0\) (d) The set of all polynomials in \(P_{4}\) having at least one real root

Given \(\mathbf{x}_{1}=(1,1,1)^{T}\) and \(\mathbf{x}_{2}=(3,-1,4)^{T}:\) (a) Do \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) span \(\mathbb{R}^{3} ?\) Explain. (b) Let \(\mathbf{x}_{3}\) be a third vector in \(\mathbb{R}^{3}\) and \(\operatorname{set} X=\) \(\left(\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right) .\) What condition \((\mathrm{s})\) would \(X\) have to satisfy in order for \(\mathbf{x}_{1}, \mathbf{x}_{2},\) and \(\mathbf{x}_{3}\) to form a basis for \(\mathbb{R}^{3}\) ? (c) Find a third vector \(\mathbf{x}_{3}\) that will extend the set \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}\right\\}\) to a basis for \(\mathbb{R}^{3}\).

In \(\mathbb{R}^{4}\), let \(U\) be the subspace of all vectors of the form \(\left(u_{1}, u_{2}, 0,0\right)^{T},\) and let \(V\) be the subspace of all vectors of the form \(\left(0, v_{2}, v_{3}, 0\right)^{T}\). What are the dimensions of \(U, V, U \cap V, U+V ?\) Find a basis for each of these four subspaces. (See Exercises 20 and \(22 \text { of Section } 2 .)\)

Let \(A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times r},\) and \(C=A B .\) Show that (a) if \(A\) and \(B\) both have linearly independent column vectors, then the column vectors of \(C\) will also be linearly independent. (b) if \(A\) and \(B\) both have linearly independent row vectors, then the row vectors of \(C\) will also be linearly independent. [Hint: Apply part (a) to \(C^{T}\).]
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