*This preview shows
pages
1–10. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **MATH 304, Fall 2011 Linear Algebra MATH 304 Linear Algebra Lecture 13: Review for Test 1. Topics for Test 1 Part I: Elementary linear algebra (Leon 1.1–1.5, 2.1–2.2) • Systems of linear equations: elementary operations, Gaussian elimination, back substitution. • Matrix of coefficients and augmented matrix. Elementary row operations, row echelon form and reduced row echelon form. • Matrix algebra. Inverse matrix. • Determinants: explicit formulas for 2 × 2 and 3 × 3 matrices, row and column expansions, elementary row and column operations. Topics for Test 1 Part II: Abstract linear algebra (Leon 3.1–3.4, 3.6) • Vector spaces (vectors, matrices, polynomials, functional spaces). • Subspaces. Nullspace, column space, and row space of a matrix. • Span, spanning set. Linear independence. • Bases and dimension. • Rank and nullity of a matrix. Sample problems for Test 1 Problem 1 (7 pts.) Find a quadratic polynomial p ( x ) such that p (1) = 1, p (2) = 3, and p (3) = 7. Problem 2 (15 pts.) Let A = 1 − 2 4 1 2 3 2 0 2 − 1 1 2 0 1 . (i) Evaluate the determinant of the matrix A . (ii) Find the inverse matrix A − 1 . Problem 3 (12 pts.) Determine which of the following subsets of R 3 are subspaces. Briefly explain. (i) The set S 1 of vectors ( x , y , z ) ∈ R 3 such that xyz = 0. (ii) The set S 2 of vectors ( x , y , z ) ∈ R 3 such that x + y + z = 0. (iii) The set S 3 of vectors ( x , y , z ) ∈ R 3 such that y 2 + z 2 = 0. (iv) The set S 4 of vectors ( x , y , z ) ∈ R 3 such that y 2 − z 2 = 0. Problem 4 (16 pts.) Let B = − 1 4 1 1 1 2 − 1 − 3 − 1 2 − 1 1 . (i) Find the rank and the nullity of the matrix B . (ii) Find a basis for the row space of B , then extend this basis to a basis for R 4 . (iii) Find a basis for the nullspace of B . Bonus Problem 5 (8 pts.) Show that the functions f 1 ( x ) = x , f 2 ( x ) = xe x , and f 3 ( x ) = e − x are linearly independent in the vector space C ∞ ( R ). Bonus Problem 6 (15 pts.) Let V be a finite-dimensional vector space and V be a proper subspace of V (where proper means that V negationslash = V ). Prove that dim V < dim V . Problem 1. Find a quadratic polynomial p ( x ) such that p (1) = 1, p (2) = 3, and p (3) = 7. Let p ( x ) = ax 2 + bx + c . Then p (1) = a + b + c , p (2) = 4 a + 2 b + c , and p (3) = 9 a + 3 b + c . The coefficients a , b , and c have to be chosen so that a + b + c = 1 , 4 a + 2 b + c = 3 , 9 a + 3 b + c = 7 . We solve this system of linear equations using elementary operations: a + b + c = 1 4 a + 2 b + c = 3 9 a + 3 b + c = 7 ⇐⇒ a + b + c = 1 3 a + b = 2 9 a + 3 b + c = 7 ⇐⇒ a + b + c = 1 3 a + b = 2 9 a + 3 b + c = 7 ⇐⇒ a + b + c = 1 3 a + b = 2 8 a + 2 b = 6 ⇐⇒ a + b + c = 1 3 a + b = 2 4 a + b = 3 ⇐⇒ a + b + c = 1 3 a + b = 2 a = 1 ⇐⇒ a + b + c = 1 b =...

View
Full
Document