Solutions to Math 51 First Exam — April 21, 2011
... (4 points) We’ve seen that one basis for C(A) is formed by taking the columns of A that correspond to the pivot-columns of rref(A). Thus, {a1 , a2 , a4 } is a basis for C(A). From this, it immediately follows that dim C(A) = 3 and therefore no set of size 1, 2, or 4 can ever be a basis for C(A); we ...
... (4 points) We’ve seen that one basis for C(A) is formed by taking the columns of A that correspond to the pivot-columns of rref(A). Thus, {a1 , a2 , a4 } is a basis for C(A). From this, it immediately follows that dim C(A) = 3 and therefore no set of size 1, 2, or 4 can ever be a basis for C(A); we ...
Ch 16 Geometric Transformations and Vectors Combined Version 2
... Ex 1. Rotate the vector w = <3,-2> by 90 . What is the component form of the resulting vector? ...
... Ex 1. Rotate the vector w = <3,-2> by 90 . What is the component form of the resulting vector? ...
Applied Math 9 are two ways to describe a line. If the line is not
... f(x1 ; x2 ; x3) : (x1 ; x2 ; x3 ) = (v1 ; v2 ; v3 )t + (z1 ; z2 ; z3 ) for some real number tg is the line through (z1 ; z2 ; z3 ) that points in the direction (v1 ; v2 ; v3 ). (We can also consider the line as a function of the independent variable t and with the dependent variables x1 ; x2 ; and x ...
... f(x1 ; x2 ; x3) : (x1 ; x2 ; x3 ) = (v1 ; v2 ; v3 )t + (z1 ; z2 ; z3 ) for some real number tg is the line through (z1 ; z2 ; z3 ) that points in the direction (v1 ; v2 ; v3 ). (We can also consider the line as a function of the independent variable t and with the dependent variables x1 ; x2 ; and x ...
Summary: Orthogonal Functions 1. Let C0(a, b) denote the space of
... (b) The vectors are mutually orthogonal. That is too say, (fi , fj ) = 0 for i 6= j, for all i, j = 1, 2, 3, . . .. Furthermore, if each vector fi ∈ B has norm 1, then the collection B is an orthonormal set of vectors. p ...
... (b) The vectors are mutually orthogonal. That is too say, (fi , fj ) = 0 for i 6= j, for all i, j = 1, 2, 3, . . .. Furthermore, if each vector fi ∈ B has norm 1, then the collection B is an orthonormal set of vectors. p ...
4.3 COORDINATES IN A LINEAR SPACE By introducing
... By introducing coordinates, we can transform any n-dimensional linear space into Rn 4.3.1 Coordinates in a linear space Consider a linear space V with a basis B consisting of f1, f2, ...fn. Then any element f of V can be written uniquely as f = c1f1 + c2f2 + · · · + cnfn, for some scalars c1, c2, .. ...
... By introducing coordinates, we can transform any n-dimensional linear space into Rn 4.3.1 Coordinates in a linear space Consider a linear space V with a basis B consisting of f1, f2, ...fn. Then any element f of V can be written uniquely as f = c1f1 + c2f2 + · · · + cnfn, for some scalars c1, c2, .. ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.