Math Problem Statement

Olkoon A, B ∈ ℂⁿˣⁿ. Osoita, että joukko R = {x ∈ ℂⁿ | Ax = Bx} on vektoriavaruuden ℂⁿ aliavaruus.

Solution

The problem asks you to show that the set R={xCnAx=Bx}R = \{ x \in \mathbb{C}^n \mid Ax = Bx \} is a subspace of the vector space Cn\mathbb{C}^n, where AA and BB are square matrices of size n×nn \times n.

To prove that RR is a subspace, we need to check the following properties:

  1. Zero vector: Check that the zero vector x=0x = 0 belongs to the set RR.
  2. Closed under addition: If x1x_1 and x2x_2 are elements of RR, then x1+x2x_1 + x_2 must also be an element of RR.
  3. Closed under scalar multiplication: If xRx \in R and cCc \in \mathbb{C}, then cxcx must also be an element of RR.

Let me now walk through each of these steps.

  1. Zero vector:

    • We check if x=0x = 0 satisfies Ax=BxAx = Bx.
    • If x=0x = 0, then A(0)=0A(0) = 0 and B(0)=0B(0) = 0, so A0=B0A0 = B0 holds true.
  2. Closed under addition:

    • Let x1x_1 and x2x_2 be in RR. This means Ax1=Bx1A x_1 = B x_1 and Ax2=Bx2A x_2 = B x_2.
    • Consider x1+x2x_1 + x_2. We want to show A(x1+x2)=B(x1+x2)A(x_1 + x_2) = B(x_1 + x_2).
    • Using linearity of AA and BB, we have: A(x1+x2)=Ax1+Ax2=Bx1+Bx2=B(x1+x2).A(x_1 + x_2) = A x_1 + A x_2 = B x_1 + B x_2 = B(x_1 + x_2).
    • Therefore, x1+x2Rx_1 + x_2 \in R.
  3. Closed under scalar multiplication:

    • Let xRx \in R, so Ax=BxA x = B x, and let cCc \in \mathbb{C}.
    • Consider cxc x. We want to show A(cx)=B(cx)A(c x) = B(c x).
    • Using linearity of AA and BB, we have: A(cx)=cAx=cBx=B(cx).A(c x) = c A x = c B x = B(c x).
    • Therefore, cxRc x \in R.

Since all three properties hold, the set RR is indeed a subspace of Cn\mathbb{C}^n.

Do you need more details or clarifications on any of the steps? Here's a list of further questions that could help you dive deeper into the topic:

  1. What is the rank of the matrix ABA - B, and how does it relate to the dimension of RR?
  2. How do we define the kernel of a matrix, and how is it connected to subspaces like RR?
  3. How can we prove that a linear map has a nontrivial kernel under certain conditions?
  4. What would happen if AA and BB were not square matrices? How would this affect the subspace argument?
  5. How can we apply this proof to analyze similar problems with other conditions on the matrices?

Tip: Remember that to prove something is a subspace, you always check the three essential properties: containing the zero vector, being closed under addition, and being closed under scalar multiplication.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Subspaces
Complex Vector Spaces

Formulas

R = {x ∈ ℂⁿ | Ax = Bx}

Theorems

Subspace Criteria: Zero Vector, Closure under Addition, Closure under Scalar Multiplication

Suitable Grade Level

Undergraduate Linear Algebra