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单选题设是数域P上n维线性空间上的线性变换,则

A. 若是可逆变换,则它在任何一组基下的矩阵都是可逆矩阵

B. 若线性无关,则也线性无关

C. 若在某组基下的矩阵是对角矩阵,则有n个互不相同的特征值

D. 必定存在一组基,使得在这组基下的矩阵是对角矩阵

高等代数课程封面

学科：高等代数

时间：2023-06-17 16:00:25

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数域P上全体n阶方阵组成的线性空间V上的两个线性变换<img src="https://tihai-oss-cloud.itihey.com/img/d3ded8acfa9994a8519ad3b2c005bb95.png">、<img src="https://tihai-oss-cloud.itihey.com/img/3856c537885c466f12bf6680f4b4e1e8.png">定义为:<img src="https://tihai-oss-cloud.itihey.com/img/1bd966a7532f3143f358cd1d39d260dc.png">,<img src="https://tihai-oss-cloud.itihey.com/img/1a5c59e37365e44643568927cadad3e0.png">,<img src="https://tihai-oss-cloud.itihey.com/img/97c0822987140e31a93e7bd2e89bb362.png">,则<img src="https://tihai-oss-cloud.itihey.com/img/635a644b8f07bbe53131121e12665414.png">为

A. <img src="https://tihai-oss-cloud.itihey.com/img/275a207fb4538c43036333e99313f9e7.png">B. <img src="https://tihai-oss-cloud.itihey.com/img/d1459ff36e6a037b02c3d9dae4cd6e1c.png">C. <img src="https://tihai-oss-cloud.itihey.com/img/0561c2500e536c0f6634ef4e304ff951.png">D. <img src="https://tihai-oss-cloud.itihey.com/img/f5f489d2264239d589532034811a9973.png">

下列各变换<img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png">是<img src="https://tihai-oss-cloud.itihey.com/img/6ecde00f734004019f53ccac72669b86.webp">上的线性变换的是

A. <img src="https://tihai-oss-cloud.itihey.com/img/763cd66dd27668376d39809ea99e5c0d.png">B. <img src="https://tihai-oss-cloud.itihey.com/img/193d74dc20847ef7e660cedbdeb1116c.png">C. <img src="https://tihai-oss-cloud.itihey.com/img/69c32054d2bbcbc2c28427d458913bcf.png">D. <img src="https://tihai-oss-cloud.itihey.com/img/210637a22672754e705404ea3be3d969.png">

设<img src="https://tihai-oss-cloud.itihey.com/img/322b5636a4a70ceda11e1a01dcde9f22.png">是线性空间V上的一个线性变换,<img src="https://tihai-oss-cloud.itihey.com/img/749c8b45470df36bb50f86259f654cbc.png">、<img src="https://tihai-oss-cloud.itihey.com/img/bc413a7e7df4e2a18ec520baa0569123.png">是<img src="https://tihai-oss-cloud.itihey.com/img/d3ded8acfa9994a8519ad3b2c005bb95.png">的两个不同特征根,<img src="https://tihai-oss-cloud.itihey.com/img/33a0088c8252985f9bec3c7894a924f7.png">、<img src="https://tihai-oss-cloud.itihey.com/img/1a62b9f8c0c3320901d70a17acee7bed.png">分别为<img src="https://tihai-oss-cloud.itihey.com/img/d3ded8acfa9994a8519ad3b2c005bb95.png">的属于<img src="https://tihai-oss-cloud.itihey.com/img/749c8b45470df36bb50f86259f654cbc.png">、<img src="https://tihai-oss-cloud.itihey.com/img/bc413a7e7df4e2a18ec520baa0569123.png">的特征向量,则

A. <img src="https://tihai-oss-cloud.itihey.com/img/c38b898e2be2df64fc893d131ffa6b3f.png">是<img src="https://tihai-oss-cloud.itihey.com/img/322b5636a4a70ceda11e1a01dcde9f22.png">的特征根B. <img src="https://tihai-oss-cloud.itihey.com/img/cc4c69030ae7a42a43ded04361a6e0ba.png">是<img src="https://tihai-oss-cloud.itihey.com/img/322b5636a4a70ceda11e1a01dcde9f22.png">的特征向量C. <img src="https://tihai-oss-cloud.itihey.com/img/33a0088c8252985f9bec3c7894a924f7.png">与<img src="https://tihai-oss-cloud.itihey.com/img/1a62b9f8c0c3320901d70a17acee7bed.png">线性无关D. <img src="https://tihai-oss-cloud.itihey.com/img/33a0088c8252985f9bec3c7894a924f7.png">与<img src="https://tihai-oss-cloud.itihey.com/img/1a62b9f8c0c3320901d70a17acee7bed.png">线性相关

设可逆矩阵A一个特征值为2,则<img src="https://tihai-oss-cloud.itihey.com/img/59cbb632f5fa28ffe0c1539404a7dbc9.webp">有一个特征值为

A. <img src="https://tihai-oss-cloud.itihey.com/img/1985eaeb8087824b39eb9af95ceb88c1.webp">B. <img src="https://tihai-oss-cloud.itihey.com/img/969ccf3b9e8784ca389a6f514e60d21a.webp">C. <img src="https://tihai-oss-cloud.itihey.com/img/53728ce2aa2dc2065d5d97d3260fdf2f.webp">D. <img src="https://tihai-oss-cloud.itihey.com/img/c5b1d50f24f2a5ca751c5e6d0bfb3d9d.webp">

n阶矩阵A与对角矩阵相似的充要条件是

A. A有n个线性无关的特征向量B. A可逆C. A的n个特征值互不相同D. A无零特征值

设 <img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png"> 是数域P上n维线性空间<img src="https://tihai-oss-cloud.itihey.com/img/43cba1a756cf73262150be4dcc55b5d7.png">上的线性变换,则

A. 若 <img src="https://tihai-oss-cloud.itihey.com/img/a57605141c2ccbef0a47678e1435c41e.png">是可逆变换,则它在任何一组基下的矩阵都是可逆矩阵B. 若 <img src="https://tihai-oss-cloud.itihey.com/img/b30f0e20c5dc0d3a84f2c372a2431c7a.png">线性无关,则 <img src="https://tihai-oss-cloud.itihey.com/img/c793f152d14e1c04d949f9dbdfa2af10.png">也线性无关C. 若 <img src="https://tihai-oss-cloud.itihey.com/img/14d672af777be80510d59769e2939f1e.png">在某组基下的矩阵是对角矩阵,则 <img src="https://tihai-oss-cloud.itihey.com/img/88bbd8e4291ffe260fd1ec5a3decb3f4.png">有n个互不相同的特征值D. 必定存在一组基,使得 <img src="https://tihai-oss-cloud.itihey.com/img/14d672af777be80510d59769e2939f1e.png">在这组基下的矩阵是对角矩阵

设 σ 是数域P上n维线性空间V上的线性变换,下列说法错误的是

A. 设σ在某组基下的矩阵是<img src="https://tihai-oss-cloud.itihey.com/img/152b35092201fc3a9190bff9ff1f9819.png">,且<img src="https://tihai-oss-cloud.itihey.com/img/af3c58ee737e716089cbfc1fa238f031.png">是<img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png">的特征值,则线性方程组<img src="https://tihai-oss-cloud.itihey.com/img/8ddea108ed9d58326365bdda0d2ca3f0.png">的解空间就是不变子空间<img src="https://tihai-oss-cloud.itihey.com/img/fe240e9efdda30f3f855162ee7e1c45d.png">B. 在σ在某组基下的矩阵是<img src="https://tihai-oss-cloud.itihey.com/img/152b35092201fc3a9190bff9ff1f9819.png">,若线性方程组<img src="https://tihai-oss-cloud.itihey.com/img/7d2d2780e141e897f6b37313a7cc60bc.png">有非零解,则 <img src="https://tihai-oss-cloud.itihey.com/img/7f22ee000dbad73ab3d437c1fedc36ca.png">是σ的特征值C. 若<img src="https://tihai-oss-cloud.itihey.com/img/bb1e7a5215cec84730236e22267d34b8.png">是<img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png">的特征值,则特征子空间<img src="https://tihai-oss-cloud.itihey.com/img/fe240e9efdda30f3f855162ee7e1c45d.png">必定不是零空间D. 若V的所有特征子空间的和等于V,则必定存在V的一组基使得 σ 在这组基下的矩阵是对角矩阵

设<img src="https://tihai-oss-cloud.itihey.com/img/c776dcf3d08e7d1a9b8f5afb0581ff18.png">是数域P上线性空间V的线性变换,<img src="https://tihai-oss-cloud.itihey.com/img/a469291f406a0af57d746c09342aea6c.png">和<img src="https://tihai-oss-cloud.itihey.com/img/186c79272a5ebcf9a767f3a88aa8c747.png">是<img src="https://tihai-oss-cloud.itihey.com/img/7036e5d34b6d61f2389d765d50152ecd.png">的分别属于特征值<img src="https://tihai-oss-cloud.itihey.com/img/bf39ff2aeb947424e458ca9119a62c33.png">的特征向量,那么

A. 若 <img src="https://tihai-oss-cloud.itihey.com/img/6e619f0c9e9337326f65d41175611753.png">和 <img src="https://tihai-oss-cloud.itihey.com/img/186c79272a5ebcf9a767f3a88aa8c747.png">线性相关,则 <img src="https://tihai-oss-cloud.itihey.com/img/61987e2f03f12f0463ecbbf404ec8df1.png">B. 若 <img src="https://tihai-oss-cloud.itihey.com/img/b2fa6015685f554ff06e2e500d8ab9ea.png">,则 <img src="https://tihai-oss-cloud.itihey.com/img/a469291f406a0af57d746c09342aea6c.png">和 <img src="https://tihai-oss-cloud.itihey.com/img/186c79272a5ebcf9a767f3a88aa8c747.png">线性相关C. 若 <img src="https://tihai-oss-cloud.itihey.com/img/a469291f406a0af57d746c09342aea6c.png">和 <img src="https://tihai-oss-cloud.itihey.com/img/186c79272a5ebcf9a767f3a88aa8c747.png">线性无关,则 <img src="https://tihai-oss-cloud.itihey.com/img/61987e2f03f12f0463ecbbf404ec8df1.png">D. 若 <img src="https://tihai-oss-cloud.itihey.com/img/61987e2f03f12f0463ecbbf404ec8df1.png">,则 <img src="https://tihai-oss-cloud.itihey.com/img/a469291f406a0af57d746c09342aea6c.png">和 <img src="https://tihai-oss-cloud.itihey.com/img/186c79272a5ebcf9a767f3a88aa8c747.png">线性无关

线性变换<img src="https://tihai-oss-cloud.itihey.com/img/70fb25c01ef05b0676fceabc95a9e12e.png">: <img src="https://tihai-oss-cloud.itihey.com/img/17332c3297fcd694e1c9391174eca741.png">下的矩阵是

A. <img src="https://tihai-oss-cloud.itihey.com/img/81a5d2200c6a23fedab7676810a20be0.png">B. (<img src="https://tihai-oss-cloud.itihey.com/img/1a93822960da871be69faadfa7c654dc.png">)C. (<img src="https://tihai-oss-cloud.itihey.com/img/2b0bd1a580bf7402bd21d2097cbd8e49.png">)D. (<img src="https://tihai-oss-cloud.itihey.com/img/8a0eee8e63d70ca87e14f04cc86f310d.png">)

设<img src="https://tihai-oss-cloud.itihey.com/img/c871d84fb6ab25f5619e1bf7350e96be.png">是分别属于<img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png">的两两不同的特征根<img src="https://tihai-oss-cloud.itihey.com/img/67a5bb0cec48b1388f37214b2697e93a.png">的特征向量.其中<img src="https://tihai-oss-cloud.itihey.com/img/5b1d066a441a241b5b8aa137b467d181.png">V是向量空间,则下列结论正确的是

A. <img src="https://tihai-oss-cloud.itihey.com/img/39d1ffa1b5ab40ea56c1bc71e10d8f38.png">线性无关B. <img src="https://tihai-oss-cloud.itihey.com/img/f316a07fb470530a31cb2d10f24ccb34.png">C. <img src="https://tihai-oss-cloud.itihey.com/img/a30e9e43661f8bc140c53aa049c066dc.png">是 <img src="https://tihai-oss-cloud.itihey.com/img/b8de5dc0971b506ab85e4f7c84d9bd8d.png">D. <img src="https://tihai-oss-cloud.itihey.com/img/7e2fa24a0525b5e6bc2a7884d67dc2b6.png">是 <img src="https://tihai-oss-cloud.itihey.com/img/61f515f42030a60730d09ca23ab00c03.png">的特征向量

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