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Chapter 5: Problem 60

Suppose \(F: V \rightarrow U\) is linear. Show that \(F(-v)=-F(v).\)

Short Answer

Expert verified
By applying the properties of a linear transformation, we have shown that for any vector \(v\) in vector space \(V\), \(F(-v) = -F(v)\) when \(F: V \rightarrow U\) is a linear transformation. This is proven by using the scalar property of linear transformations, with the scalar being -1.

Step by step solution

01

Recall the properties of a linear transformation

In order to show that F(-v) = -F(v), we must use the fact that F is a linear transformation. Recall that a function F from vector space V to U is linear if it satisfies the following two properties for all vectors x and y in V, and all scalars c: 1. F(x + y) = F(x) + F(y) 2. F(c * x) = c * F(x)
02

Apply the scalar property to -1 times v

Now, let's apply the linear transformation F to the vector -v. Using the second property of linear transformations, we have: F(-1 * v) = -1 * F(v)
03

Relate the result back to our goal expression

Recall that our goal is to show that F(-v) = -F(v). We can rewrite -v as -1 times v. So our goal expression F(-v) can be rewritten as F(-1 * v). We've already found that F(-1 * v) is equal to -1 times F(v), or -F(v). So we have reached our goal expression: F(-v) = -F(v). This demonstrates that for a linear transformation F, it holds that F(-v) = -F(v) for any vector v in V.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Linear Transformations
Linear transformations are a fundamental concept in linear algebra. They bridge the gap between vector spaces and function theory, allowing operations on vectors to be studied through algebraic structures.
For a function
  • If a function \( F: V \to U \) is linear, it follows two main properties: \( F(x + y) = F(x) + F(y) \) and \( F(c \cdot x) = c \cdot F(x) \) for all vectors \( x, y \in V \) and scalars \( c \).
  • The idea is that linearity maintains the structure and relationships of the original vector space after transformation. This means the way vectors add and how scalars multiply them are preserved under \( F \).
  • These properties make linear functions predictable and useful in various applications, such as computer graphics, where they can simplify complex transformations.
With these properties, we can identify and verify whether a transformation is truly linear by testing these two conditions.
Scalar Multiplication in Linear Algebra
Scalar multiplication is a process in linear algebra where each element of a vector is multiplied by a scalar (a real number). This operation stretches or shrinks the vector while maintaining its original direction unless the scalar is negative, which also reverses the direction.
In the context of linear transformations:
  • The property \( F(c \cdot x) = c \cdot F(x) \) indicates how scalar multiplication interacts with linear transformations. It shows that the transformation scales both the vector and its resulting vector by the same amount.
  • In our exercise, this understanding is crucial because it allows the transformation of any scaled version of a vector, including \(-1 \times v\), leading directly to \(-F(v)\).
  • The effect of scalar multiplication in this case reinforces the linearity of the transformation, ensuring the transformation's consistency and applicability.
Vector Spaces
Vector spaces are the foundational structure within which linear algebra operates. A vector space consists of a set of vectors where two operations are defined: vector addition and scalar multiplication. These operations follow specific rules that maintain the space's structure.
Within vector spaces:
  • Vectors can be added together, or multiplied by scalars, satisfying particular properties like associativity, commutativity, and distributivity, which allow complex operations to be simplified.
  • A vector space must also include a zero vector (an additive identity) and each vector must have an additive inverse (like \( -v \) for vector \( v \)).
  • This concept of additive inverse is vital in proving statements involving linear transformations like \( F(-v) = -F(v) \) because it fundamentally relies on negating elements within the space.
Thus, understanding vector spaces lays the groundwork for comprehending how linear transformations like scalar multiplication and vector addition operate systematically and effectively within this conceptual framework.

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Most popular questions from this chapter

Let \(T\) be a linear operator on a finite-dimensional vector space \(V\) with the distinct eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) and corresponding multiplicities \(m_{1}, m_{2}, \ldots, m_{k}\). Suppose that \(\beta\) is a basis for \(\mathrm{V}\) such that \([\mathrm{T}]_{\beta}\) is an upper triangular matrix. Prove that the diagonal entries of \([T]_{\beta}\) are \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) and that each \(\lambda_{i}\) occurs \(m_{i}\) times \((1 \leq i \leq k)\).

Find the dimension \(d\) of \((a) \operatorname{Hom}\left(\mathbf{R}^{2}, \mathbf{R}^{8}\right),(b) \operatorname{Hom}\left(\mathbf{P}_{4}(t), \mathbf{R}^{3}\right),(\mathrm{c}) \operatorname{Hom}\left(\mathbf{M}_{2,4}, \mathbf{P}_{2}(t)\right)\)

Suppose \(F\) and \(G\) are linear operators on \(V\) and that \(F\) is nonsingular. Assume that \(V\) has finite dimension. Show that \(\operatorname{rank}(F G)=\operatorname{rank}(G F)=\operatorname{rank}(G)\)

5.90. Let \(F: V \rightarrow U\) be linear and let \(W\) be a subspace of \(V\). The restriction of \(F\) to \(W\) is the \(\operatorname{map} F | W: W \rightarrow U\) defined by \(F | W(v)=F(v)\) for every \(v\) in \(W\). Prove the following: (a) \(F | W\) is linear; (b) \(\operatorname{Ker}(F | W)=(\operatorname{Ker} F) \cap W\) (c) \(\operatorname{Im}(F | W)=F(W)\)

Let \(F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}\) be the linear mapping for which \(F(1,2)=(2,3)\) and \(F(0,1)=(1,4) .[\) Note that \(\left.\\{(1,2),(0,1)\\} \text { is a basis of } \mathbf{R}^{2}, \text { so such a linear map } F \text { exists and is unique by Theorem } 5.2 .\right]\) Find a formula for \(F ;\) that is, find \(F(a, b).\)
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