Two subsets
Let $val6 be a vector space. We have two subsets of $val6,
and
, having respectively $val9 and $val10 elements. Answer: - If $val25, then
.
- If $val26, then
.
Two subsets II
Let $val6 be a vector space. We have two subsets of $val6,
and
, having respectively $val17 and $val18 elements. Answer: If $val37, is it true that $val40 ? |
|
If $val38, is it true that $val41 ? |
|
Dim matrix antisym
What is the dimension of the (real) vector space composed of real antisymmetric matrices of size $val6×$val6?
Dim matrix sym
What is the dimension of the (real) vector space composed of symmetric real matrices of size $val6×$val6?
Dim matrix triang
What is the dimension of the (real) vector space composed of real $val8 triangular matrices of size $val6×$val6?
Dim poly with roots
What is the dimension of the vector space composed of real polynomials of degree at most $val6, having $val8 as a root of multiplicity at least $val7?
Parametrized vector
Let v1=($val23) and v2=($val24) be two vectors in
. Find the value for the parameter t such that the vector v=($val21) belongs to the subspace of
generated by v1 and v2.
Shelf of bookshop 3 authors
A bookshop ranges its shelf of novels. - If one shows $val7 (resp. $val8, $val9) copies of each title of author A (resp. author B, author C), there will be $val16 books on the shelf.
- If one shows $val10 (resp. $val11, $val12) copies of each title of author A (resp. author B, author C), there will be $val17 books on the shelf.
How many titles are there in total for these three authors?
Dim(ker) endomorphism
Let
be a vector space of dimension $val6, and
an endomorphism. One knows that the image of
is of dimension $val7. What is the minimum of the dimension of the kernel of
?
Dim subspace by system
Let E be a sub-vector space of R$val9 defined by a homogeneous linear system. This system is composed of $val7 equations, and the rank of the coefficient matrix of this system is equal to $val6. What is the dimension of E?
Generation and dependency
Let $val6 be a vector space of dimension $val21, and let $val7 be a set of $val25 $val26. Study the truth of the following statements.
Dim intersection of subspaces
Let
be a vector space of dimension $val6, and
,
two subspaces of
with
,
. One supposes that
and
generate
. What is the dimension of the intersection
?
Image of vector 2D
Let
be a linear map, with
,
. Compute
, where
.$m_par To give your reply, one writes
.
Image of vector 2D II
Let
be a linear map, with
,
. Compute
, where
.$m_par To give your reply, one writes
.
Image of vector 3D
Let
be a linear map, with
,
,
. Compute
, where
.$m_par To give your reply, one writes
.
Image of vector 3D II
Let
be a linear map, with
,
,
. Compute
, where
.$m_par To give your reply, one writes
.