In Exercises 4045, find the projection of v onto u. Draw asketch in Exercises 40 and 41. 45.u 3.010.332.52 , v 1.344.251.66Figure 1.39 suggests two ways in which vectorsmay be used to compute the area of a triangle.The area A of the triangle in part (a) is given byand part (b) suggests the trigonometric form of thearea of a triangle: A (We can use theidentity to find sin .)

M303 Section 2.3 Notes- Characterizations of Invertible Matrices 10-14-16 Theorem 8- Invertible Matrix Theorem (IMT)- Let be × matrix; the following are equivalent: o is invertible o = o has pivot positions in its EF o = has only trivial solution o Columns of are linearly independent o LM :ℝ → ℝ given by = is 1-1 o = has a solution for all o Columns of span ℝ o LM :ℝ → ℝ given by = is onto o There exists an × matrix such that = o There exists an × matrix such that = o is invertible −1 −1 Let and be square matrices; if = , then and both invertible and , = 1 0 −2 Ex. Using the Invertible Matrix Theorem, determine if [ 3 1 −2 ]is invertible. −5 −1 9 0 −2 o ~ EF [0 4 ] 0 0 3 o Pivot in every row and column; columns are linearly independent and span ℝ Linear map :ℝ → ℝ invertible if there ex