Linear Algebra

Inner Product Spaces Exercise 1 In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on ℝ². $ (0, 1), (2, 0) $ $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ $(- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ $ (0, 0), (0, 1) $ Exercise 2 In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on $\mathbb{R}^2$. ...

Inner Products Exercise 1 Let $\mathbb{R}^2$ have the weighted Euclidean inner product: $$ \langle u, v \rangle = 2u_1v_1 + 3u_2v_2 $$and let $u = (1, 1)$, $v = (3, 2)$, $w = (0, -1)$, and $k = 3$. Compute the stated quantities. $ \langle u,v \rangle $ $ \langle ku,w \rangle $ $ \langle u+v , w \rangle $ $ | v | $ $ d\langle u,v \rangle $ $ | u-kv | $ Exercise 2 Let $\mathbb{R}^2$ have the weighted Euclidean inner product: ...

Definition of Vector Space Exercise 1: Double Additive Inverse Prove that $ -(-v) = v $ for every $ v \in V $, where $ V $ is a vector space. Exercise 2: Zero Product Property Suppose $ a \in \mathbb{F} $ (a field), $ v \in V $, and $ av = 0 $. Prove that $ a = 0 $ or $ v = 0 $. Exercise 3: Unique Solution to Vector Equation Suppose $ v, w \in V $. Explain why there exists a unique $ x \in V $ such that $ v + 3x = w $. ...

Léisung zu dësen Exercicer sinn am Archiv op luxformel ze fannen. Complex Numbers and Vector Space Exercise 1 : Complex Number Inverse Suppose $a $ and $b $ are real numbers, not both 0. Find real numbers $c $ and $d $ such that $$ \frac{1}{a + bi} = c + di. $$Exercise 2 : Cube Root of Unity Show that $$ \frac{-1 + \sqrt{3}i}{2} $$ is a cube root of 1 (meaning that its cube equals 1). ...