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The solutions manual for exam 1 of math 205, a college-level linear algebra course. It includes problems on finding parametric vector solutions of a system of linear equations, determining the image and injectivity of a linear transformation, finding the standard matrix of a linear transformation, solving linear systems, calculating matrix inverses, and proving properties of vectors and matrices. The document also includes problems on the dot product and linear independence.
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Math 205 - Exam 1 - February 3, 2006
How can these solutions be described geometrically?
(a) Does T map R^4 onto R^3? Explain. (b) Is T one-to-one? Explain.
x 1 + hx 2 = 2 3 x 1 + 6 x 2 = k
For what value(s) of h and k (if any) does this system have
(a) no solutions? (b) a unique solution? (c) infinitely many solutions?
v =
, find v · v. How does this compare to the distance from the point (0, 0) to the point (3, 4)? (Hint: Draw a right triangle and use the Pythagorean theorem..)