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Unformatted text preview: ´ , based on the basis found. a) ² = ℝ µ ,± = ¶ · ¸ · ¹ º ½ + ¾ ¾ ¿ ¿ + ¾ À : ½, ¾, ¿ ∈ ℝ Ã · Ä · Å Math121A Sample Final Exam WQ10 4 b) g = ℝ G×G , ± = {² ∈ ℝ G×G : − ² = ² ³ } . 7) Let ´: g → µ be a linear transformation. a) Show that the null space of ´ is a subspace of g . b) Show that the range of ´ is a subspace of µ . Math121A Sample Final Exam WQ10 5 8) Let g: G ± (ℝ) → G ± (ℝ) be the transformation g²³(´)µ = ³ ¶ (´) − ³(´) . Let · = {2 + ´, 2 − ´, 1 + ´ ± } be a basis for G ± (ℝ) . Find ¸g¹ º . 9) Let g: ℝ » → ℝ » be a linear transformation with the property that g²g(´ ¼)µ = g(´ ¼) for every vector ´ ¼ ∈ ℝ » . a) Let ½ be the range of g . In other words, ½ = {g(´ ¼)u´ ¼ ∈ ℝ » } . If ¾ ¼ ∈ ½ , then what is g(¾ ¼) ? b) If ´ ¼ ∈ ℝ » , then what is g(´ ¼ − g(´ ¼)) ?...
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 Spring '08
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 Math, Linear Algebra, Vector Space

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