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Chapter 2: Problem 29

$$ \text { Prove that the subspaces }\\{0\\}, V, R(T) \text {, and } N(T) \text { are all } T \text {-invariant. } $$

Short Answer

Expert verified
In summary, we have proved that the subspaces {0}, V, R(T), and N(T) are all T-invariant by analyzing each subspace individually: 1. For the zero subspace {0}, since T is a linear transformation and T(\(\textbf{0}\)) = \(\textbf{0}\), it is T-invariant. 2. The whole vector space V is T-invariant because T transforms vectors from V to V. 3. The range R(T) is T-invariant as for any w in R(T) obtained by applying T to a vector v in V; applying T to w results in another vector in R(T). 4. The null space N(T) is T-invariant because for any n in N(T), T(n) = \(\textbf{0}\) and T(\(\textbf{0}\)) = \(\textbf{0}\), which belongs to N(T).

Step by step solution

01

Prove that the zero subspace {0} is T-invariant

Consider the zero subspace {0}. We know that this subspace contains only the zero vector. Let's take the zero vector and apply the linear transformation T to it: \[T(\textbf{0})\] Since T is a linear transformation, we know that T satisfies the following property: \[T(\textbf{0}) = \textbf{0}\] So, we can conclude that T(\(\textbf{0}\)) is in the zero subspace, making it T-invariant.
02

Prove that the whole vector space V is T-invariant

Let's now consider the whole vector space V. Let v be any vector in V. We want to show that T(v) is also in V. Since T is a linear transformation from V to V, the result of T(v) will also be in V. Therefore, the whole vector space V is T-invariant.
03

Prove that the range R(T) is T-invariant

Next, let's consider the range, or the set of all possible output vectors, of T, denoted by R(T). Let w be a vector in R(T). That means there exists a vector v in V such that: \[w = T(v)\] Now, let's apply the linear transformation T to w: \[T(w) = T(T(v))\] Since T is a linear transformation, we can find a vector u in V such that: \[T(w) = T(u)\] So, T(w) is in R(T), because it's the result of a linear transformation applied to a vector in V. Thus, the range of T, R(T), is T-invariant.
04

Prove that the null space N(T) is T-invariant

Finally, let's consider the null space N(T), which is defined as the set of all vectors in V that map to the zero-vector under the transformation T: \[N(T) = \{v \in V | T(v) = \textbf{0}\}\] Let n be a vector in N(T). We want to show that T(n) is also in N(T). We know that T(n) = \(\textbf{0}\) because n is in N(T). Since T is a linear transformation, we also know that T(\(\textbf{0}\)) = \(\textbf{0}\). Hence, T(n) is in N(T), making the null space T-invariant. In summary, we have proved that the subspaces {0}, V, R(T), and N(T) are all T-invariant.

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Key Concepts

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

Linear Transformation
At the heart of understanding T-invariant subspaces lies the concept of a linear transformation. A linear transformation, let's denote it as T, is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means, if you have two vectors, \(v_1\) and \(v_2\), and a scalar \(c\), then the transformation T will satisfy the properties:

  • \(T(v_1 + v_2) = T(v_1) + T(v_2)\)
  • \(T(c \times v_1) = c \times T(v_1)\)

These properties ensure that the structure of the vector space is maintained after the transformation. Imagine you're resizing and rotating a shape; if you do it correctly, the shape's proportions and angles remain consistent鈥攖hat's essentially what a linear transformation does in the realm of vector spaces.
Null Space N(T)
The null space of a linear transformation T, often denoted by \(N(T)\), is a subset of the domain vector space V. It contains all the vectors that, when transformed by T, result in the zero vector. Mathematically, \(N(T)\) is defined as:
\[N(T) = \{v \text{ in } V | T(v) = \textbf{0}\}\]

Understanding \(N(T)\) is crucial because it gives us insight into the 'kerne' of T鈥攖elling us about the vectors that 'disappear' or lose their distinct identity in the transformation. A simple analogy might be thinking about a black hole where anything that goes in is not seen again; in the case of \(N(T)\), vectors get transformed into the zero vector.
Range R(T)
Conversely to the null space, the range of a linear transformation, represented as \(R(T)\), consists of all the possible outputs 鈥� or where vectors in V 'end up' after being transformed by T. If we define T to be a transformation from a vector space V to \(W\), then the range is a subset of \(W\). It's described as follows:

\[R(T) = \{T(v) | v \text{ is in } V\}\]

To visualize this, you might think of T as a machine that takes raw materials (vectors in V) and transforms them into products (vectors in \(R(T)\)). Not all potential products might be made鈥攕ome raw materials might be incompatible or redundant鈥攕o the range gives us a list of what's actually being produced by T.
Vector Space V
All these concepts operate within the framework of a vector space, symbolized by V. A vector space is a collection of vectors along with two operations: vector addition and scalar multiplication, which must satisfy certain conditions known as axioms. These axioms ensure that vector spaces can behave nicely under linear transformations. Within a vector space, no matter how you stretch, shrink, or rotate a vector with a linear transformation, it remains within the vector space.

Think of vector space V like the universe in which vectors live; it's vast and bound by rules of operation that maintain order. It can be as simple as the 2D plane having all the arrows you can draw, point to point, or as complex as function spaces that underpin various areas of advanced mathematics and physics.

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Most popular questions from this chapter

Let \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\). (a) If \(\mathrm{T}(a, b, c)=(a, b, 0)\), show that \(\mathrm{T}\) is the projection on the \(x y\) plane along the \(z\)-axis. (b) Find a formula for \(\mathrm{T}(a, b, c)\), where \(\mathrm{T}\) represents the projection on the \(z\)-axis along the \(x y\)-plane. (c) If \(\mathrm{T}(a, b, c)=(a-c, b, 0)\), show that \(\mathrm{T}\) is the projection on the \(x y\)-plane along the line \(L=\\{(a, 0, a): a \in R\\}\).

Let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be a linear transformation from an \(n\)-dimensional vector space \(\vee\) to an \(m\)-dimensional vector space \(W\). Let \(\beta\) and \(\gamma\) be ordered bases for \(V\) and \(W\), respectively. Prove that \(\operatorname{rank}(T)=\operatorname{rank}\left(L_{A}\right)\) and that nullity \((\mathrm{T})=\) nullity \(\left(\mathrm{L}_{A}\right)\), where \(A=[\mathrm{T}]_{\beta}^{\gamma}\). Hint: Apply Exercise 17 to Figure \(2.2\).

$$ \mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3} \text { defined by } \mathrm{T}\left(a_{1}, a_{2}\right)=\left(a_{1}+a_{2}, 0,2 a_{1}-a_{2}\right) $$

(a) Prove that if \(A\) and \(B\) are similar \(n \times n\) matrices, then \(\operatorname{tr}(A)=\) \(\operatorname{tr}(B)\). Hint: Use Exercise 13 of Section \(2.3\). (b) How would you define the trace of a linear operator on a finitedimensional vector space? Justify that your definition is welldefined.

Prove that there is an additive function \(\mathrm{T}: R \rightarrow R\) (as defined in Exercise 38) that is not linear. Hint: Let \(\mathrm{V}\) be the set of real numbers regarded as a vector space over the field of rational numbers. By the corollary to Theorem \(1.13\) (p. 61 ), \(\mathrm{V}\) has a basis \(\beta\). Let \(x\) and \(y\) be two distinct vectors in \(\beta\), and define \(f: \beta \rightarrow \mathrm{V}\) by \(f(x)=y, f(y)=x\), and \(f(z)=z\) otherwise. By Exercise 35 , there exists a linear transformation \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) such that \(\mathrm{T}(u)=f(u)\) for all \(u \in \beta\). Then \(\mathrm{T}\) is additive, but for \(c=y / x, \mathrm{~T}(c x) \neq c \mathrm{~T}(x)\).
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