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Chapter 4: Problem 12

Prove Theorem 4.3: The intersection of any number of subspaces of \(V\) is a subspace of \(V\).

Short Answer

Expert verified
The intersection W of any number of subspaces \(W_1, W_2,..., W_n\) of V is a subspace of V if: 1. W is non-empty (contains the zero vector): Since each \(W_i\) contains the zero vector, their intersection W also contains the zero vector. 2. W is closed under addition: If u and v are in W, then u and v are in each \(W_i\), and since each \(W_i\) is closed under addition, so is W. 3. W is closed under scalar multiplication: If u is in W and c is a scalar, then u is in each \(W_i\), and since each \(W_i\) is closed under scalar multiplication, so is W. Thus, Theorem 4.3 is proven: the intersection of any number of subspaces of V is a subspace of V.

Step by step solution

01

Show that the intersection W is non-empty (contains the zero vector)

Since every subspace contains the zero vector, it follows that the zero vector is in all of the subspaces W_1, W_2,..., W_n. Therefore, the zero vector is in their intersection, meaning W is non-empty.
02

Show that W is closed under addition

Let u and v be two vectors in W. By definition of the intersection, this means that u and v are in every subspace W_1, W_2,... W_n. Since all of these subspaces are closed under addition, it means that for each i, we have u+v 鈭� W_i. Since u+v is in every subspace W_i, it must also be in their intersection, so u+v 鈭� W. This shows that W is closed under addition.
03

Show that W is closed under scalar multiplication

Let u be a vector in W, and let c be a scalar. As before, since u is in W, it is also in each subspace W_1, W_2,..., W_n. Since each of these subspaces is closed under scalar multiplication, it means that for each i, we have cu 鈭� W_i. Since cu is in every subspace W_i, it must also be in their intersection, so cu 鈭� W. This shows that W is closed under scalar multiplication. Since we have proven that the intersection W contains the zero vector, and is closed under both addition and scalar multiplication, we can conclude that W is a subspace of V. This proves Theorem 4.3.

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Most popular questions from this chapter

Determine which of the following matrices have the same row space: \\[A=\left[\begin{array}{ccc} 1 & -2 & -1 \\ 3 & -4 & 5 \end{array}\right], \quad B=\left[\begin{array}{ccc} 1 & -1 & 2 \\ 2 & 3 & -1 \end{array}\right], \quad C=\left[\begin{array}{ccc} 1 & -1 & 3 \\ 2 & -1 & 10 \\ 3 & -5 & 1 \end{array}\right]\\]

Let \(V\) be the vector space of \(n\) -square matrices over a field \(K\). Show that \(W\) is a subspace of \(V\) if \(W\) consists of all matrices \(A=\left[a_{i j}\right]\) that are (a) symmetric \(\left(A^{T}=A \text { or } a_{i j}=a_{j i}\right)\), (b) (upper) triangular, (c) diagonal, (d) scalar.

Label the following statements as true or false. (a) The function det: \(\mathrm{M}_{2 \times 2}(F) \rightarrow F\) is a linear transformation. (b) The determinant of a \(2 \times 2\) matrix is a linear function of each row of the matrix when the other row is held fixed. (c) If \(A \in \mathrm{M}_{2 \times 2}(F)\) and \(\operatorname{det}(A)=0\), then \(A\) is invertible. (d) If \(u\) and \(v\) are vectors in \(\mathrm{R}^{2}\) emanating from the origin, then the area of the parallelogram having \(u\) and \(v\) as adjacent sides is $$ \operatorname{det}\left(\begin{array}{l} u \\ v \end{array}\right) \text {. } $$ (e) A coordinate system is right-handed if and only if its orientation equals 1 .

Let \(c_{j k}\) denote the cofactor of the row \(j\), column \(k\) entry of the matrix \(A \in \mathrm{M}_{n \times n}(F)\). (a) Prove that if \(B\) is the matrix obtained from \(A\) by replacing column \(k\) by \(e_{j}\), then \(\operatorname{det}(B)=c_{j k}\). (b) Show that for \(1 \leq j \leq n\), we have $$ A\left(\begin{array}{c} c_{j 1} \\ c_{j 2} \\ \vdots \\ c_{j n} \end{array}\right)=\operatorname{det}(A) \cdot e_{j} . $$ Hint: Apply Cramer's rule to \(A x=e_{j} .\) (c) Deduce that if \(C\) is the \(n \times n\) matrix such that \(C_{i j}=c_{j i}\), then \(A C=[\operatorname{det}(A)] I\). (d) Show that if \(\operatorname{det}(A) \neq 0\), then \(A^{-1}=[\operatorname{det}(A)]^{-1} C\).

Prove that \(\operatorname{span}(S)\) is the intersection of all subspaces of \(V\) containing \(S\).
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