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

Vector Space Properties: Linear Algebra Flashcards

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Algebra - Linear Algebra

kipkogeibrxwzp Created by 9 mon ago

Cards in this deck(10)

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

For all vectors u; v and w in R^n, u + (v + w) = (u + v) + w. This means we can write u + v + w without parentheses.

For all vectors u and v in R^n, u + v = v + u

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).

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)

For all scalars c in R and all vectors u in R^n, cu is a vector in R^n

For all scalars c in R and all vectors u and v in R^n, c(u + v) = cu + cv

For all scalars c, d in R and all vectors u in R^n, (c + d)u = cu + du

1u = u

For all scalars c and d in R and all vectors u in R^n, c(du) = (cd)u

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Vector Space Properties: Linear Algebra Flashcards

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