How does technology primarily impact society? By creating new languages Through the application of scientific knowledge By reducing educational opportunities Through environmental degradation

Through the application of scientific knowledge

What is one key function of a school? To entertain students To provide structured education To be a social club To sell books

To provide structured education

Which of the following best describes happiness? A feeling of discontent A mental or emotional state of satisfaction A professional achievement A physical condition

A mental or emotional state of satisfaction

being invertible det is zero when Do elementary row operations change the row space rank(A)= for a square matrix (m = n), being surjective is equivalent to

for a square matrix (m = n), being surjective is equivalent to

n eigenvectors geom mult. of eigenvalues = alg. mult When is a matrix diagonalizable ATAx=ATb Projection onto the column space of A: Row space

When is a matrix diagonalizable

A = PDP⁻¹. When are columns indep. det(AT)= Properties of Similar Matrices A matrix is diagonalizable when it can be written as a product of three matrices:

A matrix is diagonalizable when it can be written as a product of three matrices:

A subset W of Rn is a subspace of Rn if and only if the following conditions are met: vectors in any basis for the null space is a subspace of R3 1. The zero vector is in W. 2. x + y is in W whenever x and y are in W 3. ax is in W whenever x is in W and a is any scalar number of basis vectors

1. The zero vector is in W. 2. x + y is in W whenever x and y are in W 3. ax is in W whenever x is in W and a is any scalar

If S={v1,..., vr} is a set of vectors in Rn, then the set W consisting of all linear combinations of v1,..., vr is 1. The set is linearly dependent 2. The set does not span Rn n a subspace of Rn and is called the subspace spanned by S If A is an (m × n) matrix, then n = rank(A) + nullity(A)

a subspace of Rn and is called the subspace spanned by S

Let A be an (m × n) matrix. The range of A [denoted R(A)] is the set of vectors in Rm defined by the subspace spanned by the column vectors of the matrix NO R(A) = {y: y = Ax for some x in Rn}. exactly n linearly independent vectors

R(A) = {y: y = Ax for some x in Rn}.

the range of a matrix is the same thing as the column space of that matrix. The column space is maximum number of linearly independent vectors that can exist in the space the subspace spanned by the column vectors of the matrix so that there isnt a row of the form [0 0 0 0 1] have the same dimension.

the subspace spanned by the column vectors of the matrix

the span of a set of vectors is the same thing as the range of a matrix whose columns are those vectors YES basis If A is an (m × n) matrix, then n = rank(A) + nullity(A)

the range of a matrix whose columns are those vectors

the dimension of a vector space is the basis the rank of AT NO maximum number of linearly independent vectors that can exist in the space

maximum number of linearly independent vectors that can exist in the space

For all vectors u and v in R^n, u + v is in R^n. The sum of two vectors in R^n is always a vector in R^n Distributive law two scalars Distributive law one scalar Scalar distribution The Set of all R^n is CLOSED UNDER VECTOR ADDITION

The Set of all R^n is CLOSED UNDER VECTOR ADDITION

There is a zero vector 0 in R^n such that 0+u = u for all vectors u in R^n. In fact, 0 = (0, 0, ....., 0). The Set R^n is CLOSED UNDER SCALAR MULTIPLICATION Distributive law one scalar R^n contains an ADDITIVE IDENTITY ELEMENT Each vector u in R^n contains a MULTIPLICATIVE IDENTITY ELEMBENT 1

R^n contains an ADDITIVE IDENTITY ELEMENT

For each vector u in R^n, there is a unique vector u in R^n such that u + (u) = 0. In fact, if u = (u1, u2, . . . , un) then -u = (-u1, -u2, . . . , -un) Distributive law one scalar Vector Addition is ASSOCIATIVE Each vector in R^n has a unique ADDITIVE INVERSE in R^n Distributive law two scalars

Each vector in R^n has a unique ADDITIVE INVERSE in R^n

For all scalars c in R and all vectors u in R^n, cu is a vector in R^n R^n contains an ADDITIVE IDENTITY ELEMENT The Set R^n is CLOSED UNDER SCALAR MULTIPLICATION Vector Addition is ASSOCIATIVE The Set of all R^n is CLOSED UNDER VECTOR ADDITION

The Set R^n is CLOSED UNDER SCALAR MULTIPLICATION

1u = u Distributive law one scalar Each vector u in R^n contains a MULTIPLICATIVE IDENTITY ELEMBENT 1 R^n contains an ADDITIVE IDENTITY ELEMENT The Set of all R^n is CLOSED UNDER VECTOR ADDITION

Each vector u in R^n contains a MULTIPLICATIVE IDENTITY ELEMBENT 1

the number of vectors in a basis for H - dim {0}=0 nonsingular matrix range of T linearly independent set of vectors dimension (of non-zero subspace H)

dimension (of non-zero subspace H)

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