Let \(A\) be a \(2\times 2\) matrix. Prove that if \(Av_1=\lambda_1v_1\) and \(Av_2=\lambda_2v_2\) for distinct reals \(\lambda_1\) and \(\lambda_2\) and nonzero vectors \(v_1\) and \(v_2\), then both columns of \(A-\lambda_1 I\) is a multiple of \(v_2\).
Tag Archives: linear algebra
2016-20 Finding a subspace
Let \(V_1,V_2,\ldots\) be countably many \(k\)-dimensional subspaces of \(\mathbb{R}^n\). Prove that there exists an \((n-k)\)-dimensional subspace \(W\) of \(\mathbb{R}^n\) such that \(\dim V_i\cap W=0\) for all \(i\).
2016-16 Column spaces
Let \(A\) be a square matrix with real entries such that \[ A A^T+A^T A = A+A^T.\] Prove that \(A\) and \(A^T\) have the same column space.