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Chapter 7: Problem 11

Let \(S: V \rightarrow W\) and \(T: V \rightarrow W\) be linear transformations. Given \(a\) in \(\mathbb{R},\) define functions \((S+T): V \rightarrow W\) and \((a T): V \rightarrow W\) by \((S+T)(\mathbf{v})=\) \(S(\mathbf{v})+T(\mathbf{v})\) and \((a T)(\mathbf{v})=a T(\mathbf{v})\) for all \(\mathbf{v}\) in \(V\). Show that \(S+T\) and \(a T\) are linear transformations.

Short Answer

Expert verified
Both \(S+T\) and \(aT\) are linear transformations.

Step by step solution

01

Understand the Definitions of Linear Transformations

Recall that a function \(L: V \rightarrow W\) is a linear transformation if it satisfies two properties for all \(\mathbf{u}, \mathbf{v} \in V\) and any scalar \(c \in \mathbb{R}\): \( L(\mathbf{u} + \mathbf{v}) = L(\mathbf{u}) + L(\mathbf{v}) \) (additivity) and \( L(c \mathbf{v}) = c L(\mathbf{v}) \) (homogeneity). Our goal is to show that \((S+T)\) and \((aT)\) satisfy these properties.
02

Verify Additivity for \(S+T\)

Consider two arbitrary elements \(\mathbf{u}\) and \(\mathbf{v}\) in \(V\). We need to show that \((S+T)(\mathbf{u} + \mathbf{v}) = (S+T)(\mathbf{u}) + (S+T)(\mathbf{v})\). Compute the left-hand side: \[(S+T)(\mathbf{u} + \mathbf{v}) = S(\mathbf{u} + \mathbf{v}) + T(\mathbf{u} + \mathbf{v})\]Using the linearity of \(S\) and \(T\), this becomes:\[(S(\mathbf{u}) + S(\mathbf{v})) + (T(\mathbf{u}) + T(\mathbf{v})) = (S(\mathbf{u}) + T(\mathbf{u})) + (S(\mathbf{v}) + T(\mathbf{v})) = (S+T)(\mathbf{u}) + (S+T)(\mathbf{v})\]
03

Verify Homogeneity for \(S+T\)

Consider a scalar \(c\) and an element \(\mathbf{v}\) in \(V\). We need to show that \((S+T)(c \mathbf{v}) = c (S+T)(\mathbf{v})\). Compute the left-hand side:\[(S+T)(c \mathbf{v}) = S(c \mathbf{v}) + T(c \mathbf{v})\]Applying the linearity of \(S\) and \(T\), we have:\[c S(\mathbf{v}) + c T(\mathbf{v}) = c (S(\mathbf{v}) + T(\mathbf{v})) = c (S+T)(\mathbf{v})\]
04

Verify Additivity for \(aT\)

Consider two arbitrary elements \(\mathbf{u}\) and \(\mathbf{v}\) in \(V\). We need to show that \((aT)(\mathbf{u} + \mathbf{v}) = (aT)(\mathbf{u}) + (aT)(\mathbf{v})\). Compute the left-hand side:\[(aT)(\mathbf{u} + \mathbf{v}) = a T(\mathbf{u} + \mathbf{v})\]Using the linearity of \(T\), this becomes:\[a (T(\mathbf{u}) + T(\mathbf{v})) = a T(\mathbf{u}) + a T(\mathbf{v}) = (aT)(\mathbf{u}) + (aT)(\mathbf{v})\]
05

Verify Homogeneity for \(aT\)

Consider a scalar \(c\) and an element \(\mathbf{v}\) in \(V\). We need to show that \((aT)(c \mathbf{v}) = c (aT)(\mathbf{v})\). Compute the left-hand side:\[(aT)(c \mathbf{v}) = a T(c \mathbf{v})\]Using the linearity of \(T\), this becomes:\[a (c T(\mathbf{v})) = (a c) T(\mathbf{v}) = c (a T(\mathbf{v})) = c ((aT)(\mathbf{v}))\]
06

Conclusion

We have demonstrated that both \(S+T\) and \(aT\) satisfy the properties of linearity, namely additivity and homogeneity, for all elements in \(V\) and scalars in \(\mathbb{R}\). Thus, \(S+T\) and \(aT\) are indeed linear transformations.

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Key Concepts

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

Additivity
Additivity is one of the core concepts of linear transformations. It ensures that a function behaves nicely when it comes to addition of vectors. More formally, a function or transformation is said to be additive if for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in a vector space \(V\), the transformation applied to their sum is equal to the sum of their individual transformations. In equation terms, this means that for a linear transformation \(L: V \rightarrow W\), additivity requires that \(L(\mathbf{u} + \mathbf{v}) = L(\mathbf{u}) + L(\mathbf{v})\). This property simplifies our operations and ensures that transformations don't disrupt the basic operations of vector addition. In the original exercise, we were tasked with ensuring that the transformations \(S+T\) and \(aT\) maintain this property. The solution showed how each transformation kept this structure, preserving the sum of vectors even after transformations. Without additivity, we would have unpredictable changes when adding vectors.
Homogeneity
Homogeneity is another essential property of linear transformations. It ensures that transformations respect scalar multiplication. This means that if you multiply a vector by a scalar and then apply the transformation, it is the same as applying the transformation first and then multiplying the result by the scalar. In mathematical terms, a transformation \(L: V \rightarrow W\) is homogeneous if for any scalar \(c\) and vector \(\mathbf{v}\) in \(V\), the equation \(L(c \mathbf{v}) = c L(\mathbf{v})\) holds true.The exercise required us to verify homogeneity for the transformations \(S+T\) and \(aT\). For \(S+T\), this demonstrated that multiplying a vector and transforming it gives the same result as transforming first, then scaling. Similarly, for \(aT\), applying the same principle confirmed it preserves the scale factors through transformations. Homogeneity confirms that linear transformations correctly scale vector spaces, keeping multiplication consistent.
Linearity
Linearity is a concept that combines both additivity and homogeneity. When a transformation is linear, it inherently satisfies both these properties. This means that linearity guarantees that operations within the transformation follow the normal rules of addition and scalar multiplication. A transformation \(L: V \rightarrow W\) is linear if it satisfies the following:
  • Additivity: \(L(\mathbf{u} + \mathbf{v}) = L(\mathbf{u}) + L(\mathbf{v})\) for all vectors \(\mathbf{u}, \mathbf{v}\) in \(V\).
  • Homogeneity: \(L(c \mathbf{v}) = c L(\mathbf{v})\) for all scalars \(c\) and vectors \(\mathbf{v}\) in \(V\).
In the given problem, we examined \(S+T\) and \(aT\) to confirm they are linear transformations. By proving both additivity and homogeneity for these, we affirm their linearity. This is crucial because linear systems are predictable and maintain structure across transformations. In summary, linearity ensures that transformations maintain the integrity of vector spaces, permitting consistent mathematical operations and expectations.

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Most popular questions from this chapter

Let \(U\) and \(V\) denote, respectively, the spaces of even and odd polynomials in \(\mathbf{P}_{n}\). Show that \(\operatorname{dim} U+\operatorname{dim} V=n+1 .\) [Hint: Consider \(T: \mathbf{P}_{n} \rightarrow \mathbf{P}_{n}\) where \(T[p(x)]=p(x)-p(-x) .]\)

Let \(A\) and \(B\) be matrices of size \(p \times m\) and \(n \times q .\) Assume that \(m n=p q .\) Define \(R: \mathbf{M}_{m n} \rightarrow \mathbf{M}_{p q}\) by \(R(X)=A X B\) a. Show that \(\mathbf{M}_{m n} \cong \mathbf{M}_{p q}\) by comparing dimensions. b. Show that \(R\) is a linear transformation. c. Show that if \(R\) is an isomorphism, then \(m=p\) and \(n=q .\) [Hint: Show that \(T: \mathbf{M}_{m n} \rightarrow \mathbf{M}_{p n}\) given by \(T(X)=A X\) and \(S: \mathbf{M}_{m n} \rightarrow \mathbf{M}_{m q}\) given by \(S(X)=X B\) are both one-to-one, and use the dimension theorem.]

If \(T: V \rightarrow W\) is a linear transformation, show that \(T\left(\mathbf{v}-\mathbf{v}_{1}\right)=T(\mathbf{v})-T\left(\mathbf{v}_{1}\right)\) for all \(\mathbf{v}\) and \(\mathbf{v}_{1}\) in \(V\).

In each case, (i) find a basis of ker \(T\), and (ii) find a basis of im \(T\). You may assume that \(T\) is linear. a. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{2} ; T\left(a+b x+c x^{2}\right)=(a, b)\) b. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{2} ; T(p(x))=(p(0), p(1))\) c. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} ; T(x, y, z)=(x+y, x+y, 0)\) d. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{4} ; T(x, y, z)=(x, x, y, y)\) e. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{ll}a+b & b+c \\ c+d & d+a\end{array}\right]\) f. \(T: \mathbf{M}_{22} \rightarrow \mathbb{R} ; T\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]=a+d\) g. \(T: \mathbf{P}_{n} \rightarrow \mathbb{R} ; T\left(r_{0}+r_{1} x+\cdots+r_{n} x^{n}\right)=r_{n}\) h. \(T: \mathbb{R}^{n} \rightarrow \mathbb{R} ; T\left(r_{1}, r_{2}, \ldots, r_{n}\right)=r_{1}+r_{2}+\cdots+r_{n}\) i. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; T(X)=X A-A X,\) where $$ A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] $$ j. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22} ; T(X)=X A,\) where \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]\)

\(\begin{array}{lll}\text { Exercise } & 7.3 .24 & \text { Let } V \text { consist of all sequences }\end{array}\) \(\left[x_{0}, x_{1}, x_{2}, \ldots\right)\) of numbers, and define vector operations $$ \begin{aligned} \left[x_{o}, x_{1}, \ldots\right)+\left[y_{0}, y_{1}, \ldots\right) &=\left[x_{0}+y_{0}, x_{1}+y_{1}, \ldots\right) \\ r\left[x_{0}, x_{1}, \ldots\right) &=\left[r x_{0}, r x_{1}, \ldots\right) \end{aligned} $$ a. Show that \(V\) is a vector space of infinite dimension. b. Define \(T: V \rightarrow V\) and \(S: V \rightarrow V\) by \(T\left[x_{0}, x_{1}, \ldots\right)=\left[x_{1}, x_{2}, \ldots\right)\) and \(S\left[x_{0}, x_{1}, \ldots\right)=\left[0, x_{0}, x_{1}, \ldots\right) .\) Show that \(T S=1_{V},\) so \(T S\) is one-to-one and onto, but that \(T\) is not one-to-one and \(S\) is not onto.
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