# 2018-12 Property of Eigenvectors

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 $$v_2$$.

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# 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 $$i$$.

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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.