Matricies Hw 2

The vector u results when a vector u-v is added to the vector v

True, adding the vectors algebraically gives you this.

An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1

True, a linear combination has the form c1v1+c2v2. Choosing scalars c1=1/2 and c2=0 yields (1/2)v1+0v2=(1/2)v1

The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1+x2a2+x3a3=b.

True, both statements express identical constraints on the variables x1, x2, x3.

The equation Ax=b is referred to as a vector equation

False, it is referred to as a matrix equation. A vector equation is written in linear combination form, such as x1a1+x2a2+…+xnan=b

If the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A.

True, the span of the columns of A consists of all vectors that can be written as Ax. If Ax=b has no solution/is inconsistent, b cannot be formed as a linear combination of A’s columns, meaning b is not in the span (columns of A).

If A is an mxn matrix whose columns do not span R^m, then the equation Ax=b is inconsistent for some b in R^m.

True, the columns of A span R^m iff Ax=b has a solution for every b ∈ R^m.

A homogeneous equation is always consistent

True, a homogeneous equation Ax=0 always has at least one solution, namely the trivial solution x=0, because A(0)=0

If x is a nontrivial solution of Ax=0, then every entry in x is nonzero

False, a nontrivial solution merely requires x to not be equal to 0 (at least one entry is nonzero). Other entries in x are permitted to be zero (ie. x = the column of [1 0])

The equation Ax=b is homogenous if the zero vector is a solution

True, substituting x=0 into Ax=b gives A(0)=b, which simplifies to b=0. An equation of the form Ax=0 is homogeneous by definition.

The solution set of Ax=b is obtained by translating the solution set of Ax=0.

False, this is only true when Ax=b is consistent/has one solution. If Ax=b has no solutions, its solution set is the empty set, which cannot be obtained by translating the solution set of Ax=0.

Any column of A always belongs to the span of A’s columns

True

m x n = row x column

True (# of rows x # of columns)

The columns of an mxn matrix span R^m iff every row has a pivot position in echelon form

True

A system has a non-trivial solution when…

there’s at least one free variable (column without a pivot)